---
abstract: 'This paper is focused on the cohomological aspects of Donaldson–Thomas theory for Jacobi algebras, and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman.  We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both these theorems as Poincaré-Birkhoff-Witt isomorphisms for the associated cohomological Hall algebra.  Along the way, we study the representation theory of the cohomological Hall algebra, and prove that the modules provided by the vanishing cycle cohomology of framed moduli spaces are cyclic modules, which recover the cohomological Hall algebra as a module over itself, in the limit obtained by letting the framing dimension vector become large.  This Representation-theoretic study leads naturally to a perverse filtration cohomological Hall algebras, preserved by the multiplication and comultiplication, providing explicit degenerations to universal enveloping algebras.' 
author:
- Ben Davison
- Sven Meinhardt
bibliography:
- 'Literatur.bib'
title: 'Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras'
---

Introduction
============

Background and main results
---------------------------

The critical cohomological Hall algebra $\operatorname{H}(\mathcal{A}_{B,W})$ associated with a smooth algebra $B$ and potential $W\in B/[B,B]$ was introduced by Kontsevich and Soibelman in [@KS2] as a way of categorifying the theory of Donaldson–Thomas invariants.  The meaning of “critical” here is that the Hall algebra is built out of the hypercohomology of the sheaf of vanishing cycles of the superpotential $\operatorname{Tr}(W)$ on the stack of $B$-modules.  This sheaf is supported on the critical locus of $\operatorname{Tr}(W)$, which is identified with the stack of representations of the Jacobi algebra associated to $B$ and $W$. In this way, the critical cohomological Hall algebra can be thought of as the Hall algebra categorifying Donaldson–Thomas theory for the category of representations of the Jacobi algebra. 

Donaldson–Thomas invariants themselves have by now an extensive literature, we refer to the sequence of papers by Dominic Joyce [@JoyceI], [@JoyceII], [@JoyceIII], [@JoyceIV], [@JoyceMF] for a comprehensive account, or the book [@JoyceDT] by Joyce and Song, and also to [@KS1], [@KS3] for the more general and abstract account by Kontsevich and Soibelman using motivic vanishing cycles.  In both treatments of Donaldson–Thomas theory, a key role is played by the integrality conjecture.  In this paper we describe and prove the natural categorification of the integrality conjecture in the context of the cohomological Hall algebra.  From now on we assume that $B=\mathbb{C} Q$ is the free path algebra of a quiver, and that we are given a Bridgeland stability condition $\zeta$ on the category of $\mathbb{C} Q$-modules, and that $\mu\in(-\infty,\infty]$ is a slope.  We denote by $\Lambda_{\mu}^{\zeta}\subset\mathbb{N}^{Q_0}$ the subset of dimension vectors of slope $\mu$ with respect to the stability condition $\zeta$. Via natural correspondences there is an algebra structure on the $\mathbb{N}^{Q_0}$-graded cohomology of the vanishing cycle complex on ${\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}$, the stack of $\zeta$-semistable $\mathbb{C}Q$-modules of slope $\mu$, that we denote $$\label{ua}
\operatorname{H}\left(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}\right):=\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}\operatorname{H}\left({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},\phi_{{\mathfrak{Tr}(W)}^{\zeta}_{\mu}}\otimes{\mathbb{L}}^{-\dim({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}})/2}\right).$$ These notions are properly defined and explained in Section \[Qrepsection\].  This algebra categorifies Donaldson–Thomas theory in the sense that Donaldson–Thomas invariants are obtained by considering the class in the Grothendieck group of $\mathbb{N}^{Q_0}$-graded mixed Hodge structures (equipped with a monodromy action) of the underlying mixed Hodge structure of (\[ua\]).  The main conclusion of this paper is that the introduction of the Cohomological Hall algebra structure associated to a quiver with potential should be seen as a first step towards forging a connection between the theory of refined Donaldson–Thomas invariants and the theory of quantum enveloping algebras, and that the proof of the integrality conjecture is the motivic shadow of a Poincaré–Birkhoff–Witt theorem for the associated quantum enveloping algebra.  In fact we reach this conclusion by providing all of the remaining steps, culminating in this PBW type theorem.  This programme was already articulated in [@KS2] but several components had yet to be found.  Firstly, a coproduct turning the algebra $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ into a Hopf algebra had to be defined.  A localised coproduct was found in the first author’s paper [@Da13], reinforcing the hope that $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ could be turned into a quantum enveloping algebra.  Meanwhile the integrality conjecture in the case of free path algebras of quivers with zero potential was proven by the second author and Markus Reineke in [@Meinhardt14].  In fact the proof is constructive; moreover these constructions manifestly lift to cohomology, and are not merely defined at the level of Grothendieck groups, the general setting for refined Donaldson–Thomas theory.  In addition, it is proven in [@DaMe4] that due to formal properties of the vanishing cycle functor, the methods of [@Meinhardt14] are enough to prove a very general version of the integrality conjecture, even in the presence of nonzero potential.  Similarly, the proof in [@DaMe4] is constructive, and the construction has an obvious lift to cohomology, or mixed Hodge modules, which is the lift that we explore in this paper.  For the motivic Donaldson–Thomas theory and cohomological Hall algebras associated to quivers, a vital role is always played by the grading induced by $\mathbb{N}^{Q_0}$, the semigroup of dimension vectors.  As an illustration, consider the case in which we forget about potentials and stability conditions, and consider $\operatorname{H}({\mathfrak{M}},\mathbb{Q})_{\operatorname{vir}}$, the cohomology of the stack of all finite-dimensional representations of some fixed quiver $Q$ (the subscript indicates that we normalise the constant sheaf by some system of shifts, which we define in (\[baby\])).  If $Q_0\neq \emptyset$, this cohomology is infinite-dimensional in every cohomological degree.  However if we decompose the stack according to dimension vectors $$\label{baby}
\operatorname{H}({\mathfrak{M}},\mathbb{Q}_{\operatorname{vir}})\cong\bigoplus_{\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}}\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}},\mathbb{Q}[\dim({\mathfrak{M}}_{\operatorname{\mathbf{d}}})])$$ and consider the resulting cohomology as a $\mathbb{N}^{Q_0}\times\mathbb{Z}$-graded vector space, where the extra $\mathbb{Z}$ keeps track of cohomological degree, each graded piece is finite-dimensional.  Motivic DT partition functions are expressed as formal power series in variables $q^{1/2},x_1,\ldots,x_{n}$, where for $\operatorname{\mathbf{d}}=(\operatorname{\mathbf{d}}_1,\ldots,\operatorname{\mathbf{d}}_n)$ the coefficient of $$x^{\operatorname{\mathbf{d}}}:=\prod_{i\in Q_0}x_i^{\operatorname{\mathbf{d}}_i}$$ is a formal Laurent power series in $q^{1/2}$, the characteristic function of the $\operatorname{\mathbf{d}}$th graded piece of the cohomology (\[baby\]).  These power series may have finite order poles at $q^{1/2}=0$, due to the shifts appearing in (\[baby\]).  In the general case this formal Laurent power series is replaced by a formal power series recording the finite dimensions of graded pieces of an infinite-dimensional $\mathbb{N}^{Q_0}$-graded mixed Hodge structure associated to the quiver $Q$, a potential $W$ and a stability condition $\zeta$. Furthermore, when we introduce the cohomological Hall algebra structure on this cohomology, it respects the $\mathbb{N}^{Q_0}$-grading. 

In order to make the presentation in this paper conceptually uniform, we slightly change the perspective on $\mathbb{N}^{Q_0}$-gradings.  We consider graded vector spaces (respectively, mixed Hodge structures) as sheaves of vector spaces (respectively, mixed Hodge modules) on the monoid $\mathbb{N}^{Q_0}$, considered as a scheme with infinitely many components, each isomorphic to $\operatorname{Spec}(\mathbb{C})$. Then a $\mathbb{N}^{Q_0}$-graded unital algebra is the same thing as a monoid in the category of such sheaves (respectively, mixed Hodge modules), for which the monoidal product is defined by $$\mathcal{F}\boxtimes_+\mathcal{G}:=+_*(\mathcal{F}\boxtimes \mathcal{G}),$$ where $\mathcal{F}\boxtimes\mathcal{G}$ is the usual external tensor product, and $+\colon\mathbb{N}^{Q_0}\times\mathbb{N}^{Q_0}\rightarrow \mathbb{N}^{Q_0}$ is the addition map.  I.e. a unital graded algebra structure is determined by a map $$\mathcal{F}\boxtimes_+\mathcal{F}\rightarrow \mathcal{F}$$ satisfying the obvious associativity condition, along with a unit map $\mathbb{Q}_{\{0\}}\rightarrow\mathcal{F}$ from the constant sheaf supported at the monoidal unit $0\in \mathbb{N}^{Q_0}$. The reason this makes for a more uniform presentation is that throughout the paper we consider also a relative version of the cohomological Hall algebra, defined inside the category of monodromic mixed Hodge modules on a monoid *over* $\mathbb{N}^{Q_0}$. The original, ‘absolute’ cohomological Hall algebra of [@KS2], is a monoid $$\label{absMon}
\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})\in{\mathcal{D}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$$ in the derived category of monodromic mixed Hodge modules on the space of dimension vectors for $Q$. The definition of monodromic mixed Hodge modules is recalled in Section \[MMHMs\]; for now, think of them just as an enlargement of the usual category of mixed Hodge modules. 

Instead of working with the absolute cohomological Hall algebra (\[absMon\]), and although we prove theorems regarding it, we mainly work with the monoid $$\label{intrRel}
\operatorname{\mathcal{H}}(\mathcal{A}_{W,\mu}^{\zeta})\in{\mathcal{D}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}))$$ in the category of monodromic mixed Hodge modules on the coarse moduli space of $\zeta$-semistable $\mathbb{C}Q$-modules of slope $\mu$. The monodromic mixed Hodge module (\[intrRel\]) is isomorphic to an associated graded version of the monoid (\[absMon\]) after applying the monoidal functor obtained by taking the direct image along the map $\dim\colon {\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\rightarrow\mathbb{N}^{Q_0}$, and passing to the associated graded of the perverse filtration $\operatorname{\mathfrak{P}}$ on $\operatorname{H}(\mathcal{A}^{\zeta}_{W,\mu})$, that we will come to shortly.  In symbols, there is an isomorphism of algebras $$\dim_*\operatorname{\mathcal{H}}(\mathcal{A}_{W,\mu}^{\zeta})\cong \operatorname{Gr}_{\operatorname{\mathfrak{P}}}\left(\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})\right).$$ The payoff for working in the relative setting is that we obtain more refined results, which are nevertheless easier to prove, since in this relative setting we are able to make extensive use of Morihiko Saito’s version [@Sai88; @Saito1; @Saito89] of the famous Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber [@BBD], and Saito’s theory of mixed Hodge modules in general.  The idea of proving local results on ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$ to deduce global results about Donaldson–Thomas invariants was already present, and heavily utilized, in the work of Joyce and Song, and later the proof of the integrality conjecture for path algebras in the absence of a potential by the second author and Reineke [@Meinhardt14].  This brings us to the first main result of this paper, a cohomological refinement of the integrality conjecture following the proof of the conjecture found in [@DaMe4]. 

\[ThmA\] Let $\zeta$ be a $\mu$-generic stability condition.  Define the following element of $\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}})$: $${\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}:=\begin{cases}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^\zeta_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})&\textrm{if  }{\mathcal{M}}^{\zeta\operatorname{-st}}_{\operatorname{\mathbf{d}}}\neq\emptyset \\0&\textrm{otherwise}\end{cases}$$ and let ${\mathcal{DT}}^{\zeta}_{W,\mu}$ be the direct sum of all the ${\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}$ for $\operatorname{\mathbf{d}}$ of slope $\mu$ with respect to the stability condition $\zeta$. Define $\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}=\operatorname{\mathcal{H}}(\dim_*{\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta})$ and define $\operatorname{DT}_{W,\mu}^{\zeta\operatorname{-ss}}$ to be the direct sum of all the $\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$ for $\operatorname{\mathbf{d}}$ of slope $\mu$. Then $$\operatorname{Sym}_{\boxtimes_+}\left(\operatorname{DT}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\cong \operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$$ in ${\mathcal{D}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$. Furthermore, working over the coarse moduli space of $\zeta$-semistable representations ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$, there is an isomorphism $$\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\cong\operatorname{\mathcal{H}}(\mathcal{A}_{W,\mu}^{\zeta})$$ in ${\mathcal{D}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}))$.

The definition of $\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$ is given in Section \[MMHMs\].  Requisite notions from the theory of representations of quivers, including the functors $\operatorname{Sym}_{\boxtimes_{\oplus}}$ and $\operatorname{Sym}_{\boxtimes_{+}}$, and the definition of ${\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}$, are recalled in Section \[Qrepsection\].  The functor $\operatorname{Sym}_{\boxtimes_{+}}$, via the identification between monodromic mixed Hodge modules on $\mathbb{N}^{Q_0}$ and $\mathbb{N}^{Q_0}$-graded monodromic mixed Hodge structures, is the functor that sends an $\mathbb{N}^{Q_0}$-graded monodromic mixed Hodge structure $\mathcal{F}$ to the free unital $\mathbb{N}^{Q_0}$-graded symmetric algebra generated by $\mathcal{F}$. We explain the relation to the integrality conjecture in terms of generating functions of weight polynomials, as that is the language in which the integrality conjecture is perhaps most familiar.  In the literature, this is most commonly referred to as “refined Donaldson–Thomas theory”.  Given an element $\mathcal{L}\in{\mathcal{D}^{b}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$, we define the polynomial $$\mathcal{Z}(\mathcal{L},q^{1/2},x_1,\ldots,x_n):=\sum_{\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}}\chi_q(\mathcal{L}_{\operatorname{\mathbf{d}}},q^{1/2})x^{\operatorname{\mathbf{d}}},$$ where we have used the weight polynomial $$\chi_q(\mathcal{L}_{\operatorname{\mathbf{d}}},q^{1/2}):=\sum_{i,j\in\mathbb{Z}} (-1)^i\dim(\operatorname{Gr}_W^j(\operatorname{H}^i(\mathcal{L}_{\operatorname{\mathbf{d}}})))q^{j/2}.$$ As in Theorem \[ThmA\] we fix a quiver $Q$, potential $W$, stability condition $\zeta$, and slope $\mu$. Although the total cohomology $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$ (defined as in (\[ua\])) is not in the bounded derived category ${\mathcal{D}^{b}}(\operatorname{MMHM}(\Lambda_{\mu}^{\zeta}))\subset {\mathcal{D}^{b}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$, since each $\operatorname{H}(\mathcal{A}_{W,\operatorname{\mathbf{d}}}^{\zeta})$ will typically be nonzero in arbitrarily high cohomological degree, the partition function $$\mathcal{Z}\left(\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta}),q^{1/2},x_1,\ldots,x_n\right)$$ still makes sense as a formal power series.  This is because for each $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$, and for each $j\in\mathbb{Z}$, the element $\operatorname{Gr}_W^j(\operatorname{H}^i(\mathcal{A}_{W,\operatorname{\mathbf{d}}}^{\zeta}))$ is nonzero for only finitely many values of $i$; we denote by ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))\subset {\mathcal{D}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ the full subcategory of monodromic mixed Hodge modules satisfying this condition and also the condition that for fixed $\operatorname{\mathbf{d}}$ the element $\operatorname{Gr}_W^j(\operatorname{H}(\mathcal{L}_{\operatorname{\mathbf{d}}}))$ vanishes for $j\ll 0$. Then one of the equivalent ways to define the plethystic exponential $$\operatorname{EXP}:\mathbb{Z}[q^{-1}][[q^{1/2}]][[x_1,\ldots,x_n]]_+\rightarrow\mathbb{Z}[q^{-1}][[q^{1/2}]][[x_1,\ldots,x_n]]$$ where the subscript “$+$” on the left hand side signifies that we take the subgroup of elements $\sum_{\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}}a_{\operatorname{\mathbf{d}}}(q^{1/2})x^{\operatorname{\mathbf{d}}}$ such that $a_0(q^{1/2})=0$, is via the formula for $\mathcal{L}\in{\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ $$\operatorname{EXP}\left(\mathcal{Z}(\mathcal{L},q^{1/2},x_1,\ldots,x_n)\right)=\mathcal{Z}\left(\operatorname{Sym}_{\boxtimes_+}(\mathcal{L}),q^{1/2},x_1,\ldots,x_n\right).$$ Finally, the integrality conjecture states that there exist Laurent polynomials $\Omega^{\zeta}_{W,\operatorname{\mathbf{d}}}(q^{1/2})\in\mathbb{Z}[q^{\pm 1/2}]$ such that $$\label{ics}
\mathcal{Z}(\mathcal{A}_{W,\mu}^{\zeta},q^{1/2},x_1,\ldots,x_n)=\operatorname{EXP}\left(\sum_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}} \Omega^{\zeta}_{W,\operatorname{\mathbf{d}}}(q^{1/2})x^{\operatorname{\mathbf{d}}}/(q^{1/2}-q^{-1/2})\right).$$ These $\Omega_{W,\operatorname{\mathbf{d}}}^{\zeta}(q^{1/2})$ are, by definition, the Donaldson–Thomas invariants for the slope $\mu$ and the stability condition $\zeta$. Note that $$\chi_q(\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty},\mathbb{Q})_{\operatorname{vir}},q^{1/2})=-q^{1/2}-q^{3/2}-\ldots=(q^{1/2}-q^{-1/2})^{-1}$$ and so we deduce from Theorem \[ThmA\] the equality $\Omega_{W,\operatorname{\mathbf{d}}}^{\zeta}(q^{1/2})=\chi_q(\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta},q^{1/2})$. Note also that for purely formal reasons one can always find formal Laurent power series $\Omega_{W,d}^{\zeta}(q^{1/2})$ satisfying (\[ics\]) - the content of the integrality conjecture is that these formal power series are in fact polynomials. 

We deduce the integrality conjecture from Theorem \[ThmA\]; since $\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}$ comes from a bounded complex of monodromic mixed Hodge modules, its weight polynomial is a genuine polynomial, instead of a formal function.  However, the cohomological lift provided by Theorem \[ThmA\] is much stronger than the integrality conjecture; it upgrades an equality in the Grothendieck ring of ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ to an isomorphism in that category.  This is an interesting result even before introducing the extra structure of the cohomological Hall algebra, since it shows that there is an entire theory of ‘cohomologically refined’ Donaldson–Thomas invariants waiting to be explored.  For early applications of this theory, we refer the reader to the first author’s reproof of the Kac positivity conjecture [@Da13b] (originally proved by Hausel, Letellier and Villegas [@HLRV13]) and proof of the quantum cluster positivity conjecture [@Da16a].  An advantage of considering constructions relative to the base ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$ instead of $\mathbb{N}^{Q_0}$ is that it leads naturally to the introduction of the perverse filtration on $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$. We approach this through a study of the representation theory of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$, which turns out to be governed by the vanishing cycle cohomology of moduli spaces of framed modules, as considered for example in [@Soi14] and [@Fran13].  For each framing vector ${\mathbf{f}}\in\mathbb{N}^{Q_0}$ we consider the $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$-module $$\operatorname{\mathcal{H}}(\mathcal{F}^{\zeta}_{W,{\mathbf{f}},\mu}):=\operatorname{\mathcal{H}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda^\zeta_{\mu}}\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\right)\in{\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu})),$$ where ${\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ is a shifted Tate twist of the monodromic mixed Hodge module of vanishing cycles on the moduli space ${\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ of framed $\zeta$-semistable representations with framing vector ${\mathbf{f}}$, the map $\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\colon {\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\rightarrow{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$ is the forgetful map to the coarse moduli space of $\zeta$-semistable representations, and the direct sum is over all dimension vectors of slope $\mu$. We also consider the absolute version $$\operatorname{H}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta}):=\operatorname{\mathcal{H}}\left(({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\mu}\xrightarrow{\pi^{\zeta}_{{\mathbf{f}},\mu}}{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}\xrightarrow{\dim}\Lambda^\zeta_{\mu})_*{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\right)\in{\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\Lambda^\zeta_{\mu}))$$ recovering the $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$-modules considered by Soibelman in [@Soi14].  The following theorem, which we prove in Subsection \[rep\_thm\_proof\], is the sum of our results on the representation theory of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ and $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$.

\[repthm\] Let $\mu\in (-\infty,\infty)$ be a slope, and let $\zeta$ be a not necessarily $\mu$-generic stability condition.  Each module $\operatorname{\mathcal{H}}(\mathcal{F}^{\zeta}_{W,{\mathbf{f}},\mu})$ is cyclic, generated by $\operatorname{\mathcal{H}}(\mathcal{F}_{W,{\mathbf{f}},0}^{\zeta})\cong\mathbb{Q}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_0}$. There are natural surjections $\operatorname{\mathcal{H}}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta})\rightarrow \operatorname{\mathcal{H}}(\mathcal{F}_{W,{\mathbf{f}}',\mu}^{\zeta})$ for ${\mathbf{f}}\geq{\mathbf{f}}'$, and if we fix $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$ and a cohomological degree $n$, then $\operatorname{\mathcal{H}}^n(\mathcal{F}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta})$ stabilizes, as we let ${\mathbf{f}}_i\mapsto \infty$ for all $i\in Q_0$. Furthermore, we recover $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ in the limit, considered as a left module over itself. 

Similarly, each $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$-module $\operatorname{H}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta})$ is cyclic, generated by $\operatorname{H}(\mathcal{F}_{W,{\mathbf{f}},0}^{\zeta})$, which is just $\mathbb{Q}$, placed in $\mathbb{Z}^{Q_0}$-degree zero.  For ${\mathbf{f}}\geq{\mathbf{f}}'$ there is a natural surjective map of $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$-modules $\operatorname{H}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta})\rightarrow \operatorname{H}(\mathcal{F}_{W,{\mathbf{f}}',\mu}^{\zeta})$, and as we let ${\mathbf{f}}$ tend to infinity in all arguments, $\operatorname{H}(\mathcal{F}^{\zeta}_{W,{\mathbf{f}},\mu})$ tends to $\operatorname{H}(\mathcal{A}^{\zeta}_{W,\mu})$ as a left $\operatorname{H}(\mathcal{A}^{\zeta}_{W,\mu})$-module. 

Coming back to the perverse filtration on $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$, since the forgetful map $\pi_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ is proper, we obtain perverse filtrations on $\operatorname{H}(\mathcal{F}_{W,\mu}^{\zeta})$ that give rise to a perverse filtration $\operatorname{\mathfrak{P}}$ on the limit $\operatorname{H}(\mathcal{A}^{\zeta}_{W,\mu})$ by Theorem \[repthm\].  By proving that the product and the localised coproduct on $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$ introduced in [@Da13] preserve the perverse filtration, we arrive at the following theorem regarding the cohomological Hall algebra of semistable representations of the Jacobi algebra whose dimension vectors are sent to a fixed ray in the upper half plane by a generic Bridgeland stability condition. 

\[qea\] The localised bialgebra structure on $\operatorname{H}(\mathcal{A}^{\zeta}_{W,\mu})$ induces a (non-localised) Hopf algebra structure on $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta}))$. If $\zeta$ is $\mu$-generic, this Hopf algebra is generated by its primitive elements, so that $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta}))$ is the universal enveloping algebra of the Lie algebra of its primitive elements.  There is a canonical isomorphism in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))$: $$\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\rightarrow \operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$$ that is realized via a canonical embedding ${\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\subset \operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ and the Cohomological Hall algebra multiplication.  There is a similar non-canonical isomorphism over the base $\Lambda^{\zeta}_{\mu}$ consisting of dimension vectors of slope $\mu$: $$\operatorname{Sym}_{\boxtimes_+}\left(\operatorname{DT}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\rightarrow\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta}).$$ and the map is defined via the cohomological Hall algebra product on $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$.

We finally prove a PBW theorem on the base ${\mathcal{M}}$, instead of the base ${\mathcal{M}}^{\zeta}_{\mu}$. Our result should be thought of as a cohomological lift of the following incarnation of the wall crossing formula, with $\int$ denoting the Kontsevich–Soibelman integration map, $$\label{WCI}
\int[{\mathfrak{M}},\phi_{W}]=\prod_{\infty\xrightarrow{\mu}-\infty}\int[{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu},\phi_{W}]\:\:{\large \substack{\\\\ \Huge \mathrm{=}\\ \scriptsize \substack{\mathrm{integrality}\\ \mathrm{conjecture}}}}\prod_{\infty\xrightarrow{\mu}-\infty}\sum_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}\Omega_{W,\operatorname{\mathbf{d}}}^{\zeta,\textrm{motivic}}\cdot [B\mathbb{C}^*].$$ This is considered as an identity in the motivic quantum torus of finite-dimensional modules of the Jacobi algebra for the pair $(Q,W)$. Instead of defining any of these terms we refer the reader to [@JoyceMF] and [@KS1] for more details on motivic Hall algebras and integration maps in the general theory of motivic Donaldson–Thomas invariants, or [@DaMe4] for the specific case of the motivic Hall algebra of representations of a Jacobi algebra.  In addition to categorifying the equality (\[WCI\]) to an isomorphism, we prove that this isomorphism can be realised in terms of the multiplication in the cohomological Hall algebra. 

We define $$\begin{aligned}
\operatorname{\mathcal{H}}(\mathcal{A}_W):=&\bigoplus_{\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}}\operatorname{\mathcal{H}}\left(({\mathfrak{M}}_{\operatorname{\mathbf{d}}}\rightarrow{\mathcal{M}})_*\phi_{{\mathfrak{Tr}(W)}}\otimes{\mathbb{L}}^{-\dim({\mathfrak{M}}_{\operatorname{\mathbf{d}}})/2}\right)\end{aligned}$$ where ${\mathfrak{M}}$ is the stack of all finite-dimensional $\mathbb{C}Q$-modules, and ${\mathcal{M}}$ is the coarse moduli space of all finite-dimensional $\mathbb{C}Q$-modules. 

\[strongPBW\] Let $\zeta$ be $\mu$-generic.  If $\zeta$ is $\mu$-generic for all $\mu$ there is an isomorphism $$\label{gthmd}
\boxtimes_{\oplus,\infty \xrightarrow{\mu}-\infty}^{\operatorname{tw}} \left(\operatorname{Sym}_{\boxtimes_{\oplus}}\left(q_{\mu,*}^{\zeta}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\xrightarrow{} \operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_W)$$ inside ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}))$, realized by picking an embedding of $q_{\mu,*}^{\zeta}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\subset \operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_W)$ for all slopes $\mu$, where $q_{\mu}^{\zeta}\colon {\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\rightarrow{\mathcal{M}}_{\mu}$ is the affinization map, and then using the product on $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W})$. Here the product $\boxtimes_{\oplus,\infty \xrightarrow{\mu}-\infty}^{\operatorname{tw}}$ is an ordered product, taken over descending slopes.  Similarly, there is an isomorphism $$\boxtimes_{+,\infty \xrightarrow{\mu}-\infty}^{\operatorname{tw}} \left(\operatorname{Sym}_{\boxtimes_{+}}\left(\operatorname{DT}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\xrightarrow{} \operatorname{H}(\operatorname{\mathcal{A}}_W)$$ realised by picking an embedding $\operatorname{DT}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\subset \operatorname{H}(\mathcal{A}_{W,\mu})$ for all $\mu$ and using the multiplication in the absolute version of the cohomological algebra introduced by Kontsevich and Soibelman. 

The above theorem means that there are essentially different PBW bases for different generic stability conditions, where the characteristic functions of different PBW bases are related to each other via the famous wall crossing formula of Joyce and Song, or Kontsevich and Soibelman.  The fact that different PBW bases can have different characteristic functions is explained by the definition of the monoidal product $\boxtimes^{\operatorname{tw}}_{\oplus}$ – given in Subsection \[prodss\].  The most important feature for explaining the relevance of wall crossing formulas is that in contrast with the product $\boxtimes_{\oplus}$, $\boxtimes^{\operatorname{tw}}_{\oplus}$ is not symmetric; this is the categorical analogue of the fact that the quantum torus utilized in [@KS1] as the image of the integration map from the motivic Hall algebra is generally not commutative.  In short, Theorem \[strongPBW\] contains a cohomological lift of the wall crossing formula as well as the integrality theorem. 

Acknowledgements
----------------

During the writing of this paper, Ben Davison was a postdoc at EPFL, supported by the Advanced Grant “Arithmetic and physics of Higgs moduli spaces” No.  320593 of the European Research Council.  He would like to sincerely thank Davesh Maulik for the many stimulating conversations that helped this paper along the way.  Sven Meinhardt wants to thank Tamás Hausel for giving him the opportunity to visit Lausanne and Markus Reineke for providing a wonderful atmosphere in Wuppertal to complete this research. 

How to read this paper
----------------------

The paper in its current form is somewhat long, and may seem a little technical.  It is broken roughly into three parts: there is the introduction above, a second part which goes up to Section \[cwcfSec\], and then a third part.  The purpose of the second part is to prove the cohomological integrality theorem (Theorem \[ThmA\]) and the cohomological wall crossing formula - i.e. the existence of isomorphisms as in the statement of Theorem \[strongPBW\], without reference to a cohomological Hall algebra.  For applications in cohomological Donaldson–Thomas theory and beyond, the existence of these isomorphisms often seems to be enough, and indeed this existence is already enough to deduce the integrality theorem.  The intricacy of the paper increases as we move into the third part, which is dedicated to proving that these isomorphisms can be realised as PBW isomorphisms in the cohomological Hall algebra.  Therefore the reader who cares only about integrality and wall crossing for cohomological Donaldson–Thomas invariants can safely treat this paper as ending with Section \[cwcfSec\] – hopefully this makes the paper less daunting. 

Hodge theory and equivariant vanishing cycles
=============================================

Monodromic mixed Hodge modules {#MMHMs}
------------------------------

Let $X$ be a complex variety.  Then we define as in [@Saito89; @Saito90] the category $\operatorname{MHM}(X)$ of mixed Hodge modules on $X$. See [@Saito1] for an overview of the theory.  There is an exact functor $\operatorname{rat}\colon {\mathcal{D}}(\operatorname{MHM}(X))\rightarrow{\mathcal{D}}(\operatorname{Perv}(X))$ which takes a complex of mixed Hodge modules $\mathcal{F}$ to its underlying complex of perverse sheaves, and commutes with $f_*,f_!,f^*,f^!,{\mathbb{D}}_X$ and tensor product.  If no remark is made to the contrary, and a non-derived target or domain category is not specified, all these functors, and indeed all functors for the entirety of this paper, will be considered as derived functors, even if their non-derived versions are well-defined.  Fix $X$ and let $\mathcal{B}_X$ be the full subcategory containing those $\mathcal{F}\in\operatorname{MHM}(X\times\mathbb{A}^1)$ such that for each $x\in X$, the total cohomology of $(\{x\}\times{\mathbb{G}}_m\rightarrow X\times{\mathbb{A}}^1)^*\mathcal{F}$ is an admissible variation of mixed Hodge structure on ${\mathbb{G}}_m$. Via Saito’s description of $\operatorname{MHM}(X\times\mathbb{A}^1)$, $\mathcal{B}_X\subset \operatorname{MHM}(X\times\mathbb{A}^1)$ is the full subcategory, closed under extensions, and containing the following types of objects

1.   Mixed Hodge modules ${\tilde{\mathcal{IC}}}_{Y\times \mathbb{A}^1}(\mathcal{L})[\dim(Y)+1]$, where $Y\subset X$ is an irreducible closed subvariety, and $\mathcal{L}$ is a pure variation of Hodge structure on $Y'\times\mathbb{G}_m$, for $Y'$ an open dense subvariety of $Y_{\operatorname{reg}}$

2.   Mixed Hodge modules ${\tilde{\mathcal{IC}}}_{Y\times \{0\}}(\mathcal{L})[\dim(Y)]$, for $\mathcal{L}$ a pure variation of Hodge structure on $Y'\subset Y_{\operatorname{reg}}\subset X$ as above. 

Here ${\tilde{\mathcal{IC}}}_{Y\times \mathbb{A}^1}(\mathcal{L})[\dim(Y)+1]$ and ${\tilde{\mathcal{IC}}}_{Y\times \{0\}}(\mathcal{L})[\dim(Y)]$ are the lifts of intersection complexes to mixed Hodge modules defined in [@Saito90 Thm.3.21].  Inside $\mathcal{B}_X$ there is a Serre subcategory $\mathcal{C}_X$ given by the full subcategory of $\operatorname{MHM}(X\times{\mathbb{A}}^1)$ containing those objects $\mathcal{F}$ such that the total cohomology of each $(\{x\}\times{\mathbb{A}}^1\rightarrow X\times {\mathbb{A}}^1)^*\mathcal{F}$ is an admissible variation of mixed Hodge structure on $\mathbb{A}^1$. By the rigidity theorem [@StZu85 Thm.4.20], any such mixed Hodge module is constant once restricted to the fibres $\{x\}\times\mathbb{A}^1$, and $\mathcal{C}_X$ is the essential image of the functor $\pi^*[1]\colon \operatorname{MHM}(X)\rightarrow\operatorname{MHM}(X\times\mathbb{A}^1)$, where $$\pi\colon X\times\mathbb{A}^1\rightarrow X$$ is the projection.  We define following [@KS2 Sec.7] the category $\operatorname{MMHM}(X)=\mathcal{B}_X/\mathcal{C}_X$. The natural pushforward $(X\times{\mathbb{G}}_m\rightarrow X\times{\mathbb{A}}^1)_!\colon \operatorname{MHM}(X\times{\mathbb{G}}_m)^{\operatorname{mon}}\rightarrow\operatorname{MMHM}(X)$ is an equivalence of categories, where $\operatorname{MHM}(X\times{\mathbb{G}}_m)^{\operatorname{mon}}$ is the full subcategory of $\operatorname{MHM}(X\times{\mathbb{G}}_m)$ containing those $\mathcal{F}$ such that the total cohomology of each pullback $(\{x\}\times{\mathbb{G}}_m\rightarrow X\times{\mathbb{G}}_m)^*\mathcal{F}$ is an admissible variation of mixed Hodge structure.  An explicit inverse equivalence $\Theta_X$ is provided by $$\mathcal{F}\mapsto (X\times{\mathbb{G}}_m\rightarrow X\times{\mathbb{A}}^1)^*\Psi_X\mathcal{F}.$$ where $$\Psi_X\mathcal{F}=(X\times\mathbb{A}^2\xrightarrow{\operatorname{id}\times +} X\times \mathbb{A}^1)_*\left(\mathcal{F}\boxtimes (\mathbb{G}_m\rightarrow\mathbb{A}^1)_!\mathbb{Q}_{\mathbb{G}_m}[1]\right).$$

\[adjProp\] The functor $\Psi_X$ provides a left adjoint to the natural functor $\mathcal{B}_X\rightarrow \mathcal{B}_X/\mathcal{C}_X$ which is also its right inverse. 

Let $h\in \operatorname{Hom}_{\mathcal{B}_X}(\Psi_X\mathcal{F},\mathcal{G})$. Then precomposing with the map $\nu_{\mathcal{F}}$ obtained from the following composition $$\begin{aligned}
&\mathcal{F}\xrightarrow{\cong} (X\times\mathbb{A}^2\xrightarrow{\operatorname{id}\times +} X\times \mathbb{A}^1)_*\left(\mathcal{F}\boxtimes (\{0\}\hookrightarrow\mathbb{A}^1)_!\mathbb{Q}_{\{0\}}\right)\rightarrow \\&\rightarrow (X\times\mathbb{A}^2\xrightarrow{\operatorname{id}\times +} X\times \mathbb{A}^1)_*\left(\mathcal{F}\boxtimes (\mathbb{G}_m\rightarrow\mathbb{A}^1)_!\mathbb{Q}_{\mathbb{G}_m}[1]\right)\end{aligned}$$ we obtain a map $\eta f :=f\circ\nu_{\mathcal{F}}\in\operatorname{Hom}_{\mathcal{B}_X/\mathcal{C}_X}(\mathcal{F},\mathcal{G})$, and the natural transformation of bifuntors $\eta$ can be seen to be an isomorphism on simple elements, proving the first claim.  The cone of $\tau_{\mathcal{F}}$ is isomorphic to $$(X\times\mathbb{A}^2\xrightarrow{\operatorname{id}\times +} X\times \mathbb{A}^1)_*\left(\mathcal{F}\boxtimes \mathbb{Q}_{\mathbb{A}^1}[1]\right)$$ which is zero in ${\mathcal{D}^{b}}(\mathcal{B}_X/\mathcal{C}_X)$, proving the second claim. 

It follows from Proposition \[adjProp\] (see [@Stackproject 06XM] for a proof) that the natural functor $${\mathcal{D}}(\mathcal{B}_X)/{\mathcal{D}}_{\mathcal{C}_X}(\mathcal{B}_X)\rightarrow {\mathcal{D}}(\mathcal{B}_X/\mathcal{C}_X)={\mathcal{D}}(\operatorname{MMHM}(X))$$ is an equivalence of categories, where ${\mathcal{D}}_{\mathcal{C}_X}(\mathcal{B}_X)$ is the full subcategory of ${\mathcal{D}}(\mathcal{B}_X)$ consisting of those objects whose cohomology objects lie in $\mathcal{C}_X$. The subcategory ${\mathcal{D}}_{\mathcal{C}_X}(\mathcal{B}_X)$ is stable under the Verdier duality functor ${\mathbb{D}}_{X\times{\mathbb{A}}^1}$, and so we obtain a Verdier duality functor on ${\mathcal{D}}(\operatorname{MMHM}(X))$ which we denote ${\mathbb{D}}^{\operatorname{mon}}_X$. The associated graded object $\operatorname{Gr}_W(\mathcal{F})$ of an object in $\mathcal{C}_X$ with respect to the weight filtration is also in $\mathcal{C}_X$, and so the weight filtration descends to $\operatorname{MMHM}(X)$.

If $f\colon X\rightarrow Y$ is a morphism of varieties, we define the functors $f^!,f^*,f_!,f_*$ to be the functors between the derived categories of monodromic mixed Hodge modules induced by the functors $(f\times\operatorname{id}_{\mathbb{A}^1})^!,(f\times\operatorname{id}_{\mathbb{A}^1})^*,(f\times\operatorname{id}_{\mathbb{A}^1})_!,(f\times\operatorname{id}_{\mathbb{A}^1})_*$ respectively. 

If $X$ is considered as a variety $X\xrightarrow{\tau}\mathbb{N}^{Q_0}$ over a monoid $\mathbb{N}^{Q_0}$ of dimension vectors for a quiver $Q$, we define $\operatorname{H}(X,\mathcal{F})$, for $\mathcal{F}\in{\mathcal{D}}(\operatorname{MMHM}(X))$, to be the total cohomology of $\tau_*\mathcal{F}$. So $\operatorname{H}(X,\mathcal{F})$ may be thought of as a $\mathbb{N}^{Q_0}$-graded monodromic mixed Hodge module on a point, e.g. the $\mathbb{N}^{Q_0}$-graded monodromic mixed Hodge structure underlying the total hypercohomology of $\mathcal{F}$. Similarly, we define $\operatorname{H}_c(X,\mathcal{F})$ to be the total cohomology of $\tau_!\mathcal{F}$.

The monoidal structures $\otimes$ on ${\mathcal{D}^{\geq}}(\operatorname{MMHM}(X))$ and ${\mathcal{D}^{\leq}}(\operatorname{MMHM}(X))$ are defined by $$\label{simmon}
\mathcal{F}\otimes\mathcal{G}:=(X\times {\mathbb{A}}^1\times {\mathbb{A}}^1\xrightarrow{\operatorname{id}\times +}X\times {\mathbb{A}}^1)_*(\operatorname{pr}_{1,2}^*\mathcal{F}\otimes\operatorname{pr}_{1,3}^*\mathcal{G})$$ where $\operatorname{pr}_{i,j}\colon X\times{\mathbb{A}}^1\times{\mathbb{A}}^1\rightarrow X\times{\mathbb{A}}^1$ is the projection onto the $i$th and the $j$th component. 

\[convProdDef\] More generally, let $X$ be a monoid in $\operatorname{Sch}(Y)$, the category of schemes over $Y$, with monoid map $\oplus\colon X\times_Y X\rightarrow X$. Then define $$\mathcal{F}\boxtimes_{\oplus} \mathcal{G}:=(X\times_Y X\times \mathbb{A}^1\times\mathbb{A}^1\xrightarrow{\oplus\times +} X\times\mathbb{A}^1)_*(\operatorname{pr}_{1,3}^*\mathcal{F}\otimes\operatorname{pr}_{2,4}^*\mathcal{G}).$$ We recover (\[simmon\]) in the special case in which $X$ is considered as a monoid in $\operatorname{Sch}(X)$. In addition, for $\mathcal{F}\in{\mathcal{D}^{b}}(\operatorname{MMHM}(X))$ and $\mathcal{G}\in{\mathcal{D}^{b}}(\operatorname{MMHM}(Y))$ we define $$\mathcal{F}\boxtimes \mathcal{G}:=(X\times Y\times\mathbb{A}^1\times\mathbb{A}^1\xrightarrow{\operatorname{id}_{X\times Y}\times +} X\times Y\times \mathbb{A}^1)_*(\operatorname{pr}_{1,3}^*\mathcal{F}\otimes\operatorname{pr}_{2,4}^*\mathcal{G}).$$ Then this external tensor product is biexact and preserves weight filtrations by [@KS2 Sec.4], [@Da16a Prop.3.2], or the proof of Proposition \[symmPure\] below. 

There is a fully faithful embedding $\operatorname{MHM}(X)\rightarrow \operatorname{MMHM}(X)$ given by $$i_*=(X\xrightarrow{x\mapsto (x,0)} X\times{\mathbb{A}}^1)_*$$ which is furthermore a monoidal functor, commuting with Verdier duality, as $i$ is a closed inclusion. 

Let $e\colon X\xrightarrow{x\mapsto(x,1)}X\times\mathbb{G}_m$ be the natural inclusion, then $e^*\Theta_X[-1]\colon \operatorname{MMHM}(X)\rightarrow \operatorname{MHM}(X)$ is also faithful (again using rigidity for variations of mixed Hodge structure [@StZu85 Thm.4.20]), and so we can define an exact faithful (non-derived) functor $$\label{formDef}
{\operatorname{forg}^{\operatorname{mon}}_{X}}\colon \operatorname{MMHM}(X)\rightarrow \operatorname{Perv}(X)$$ by setting ${\operatorname{forg}^{\operatorname{mon}}_{X}}:=\operatorname{forg}_Xe^*\Theta_X[-1]$, where $\operatorname{forg}_X\colon \operatorname{MHM}(X)\rightarrow\operatorname{Perv}(X)$ is the usual forgetful functor. 

Let $f$ be a regular function on a smooth algebraic variety $X$. Define $X_{<0}=f^{-1}(\mathbb{R}_{<0})$ and $X_0=f^{-1}(0)$. We define the functor $\psi_f\colon {\mathcal{D}}(\operatorname{Perv}(X))\rightarrow{\mathcal{D}}(\operatorname{Perv}(X))$ $$\psi_f:=(X_0\rightarrow X)_*(X_0\rightarrow X)^*(X_{<0}\rightarrow X)_*(X_{<0}\rightarrow X)^*$$ and define $\phi_f:=\operatorname{cone}((X_0\rightarrow X)_*(X_0\rightarrow X)^*\rightarrow\psi_f)$; this cone can indeed be made functorial – see [@KSsheaves] for details, or the proof of Proposition \[TSeq\] below for a functorial definition of $\phi_f$. The functors $\psi_f$ and $\phi_f$ have lifts to endofunctors of ${\mathcal{D}}(\operatorname{MHM}(X))$, defined by Saito.  In the sequel we consider vanishing cycles always as a functor ${\mathcal{D}}(\operatorname{MHM}(X))\rightarrow{\mathcal{D}}(\operatorname{MMHM}(X))$ via the definition of [@KS2 Def.27] $$\label{phimdef}
{\phi^{\operatorname{mon}}_{f}}:=(X\times{\mathbb{G}}_m\rightarrow X\times\mathbb{A}^1)_!\phi_{f/u}(X\times{\mathbb{G}}_m\rightarrow X)^*.$$ In (\[phimdef\]), $u$ denotes the coordinate on ${\mathbb{G}}_m$.

Let $f=0$. Then $\phi_f(\mathcal{F})\cong\mathcal{F}[1]$, since $\psi_f(\mathcal{F})=0$. On the other hand, by the same argument, ${\phi^{\operatorname{mon}}_{f}}\mathcal{F}=(X\times{\mathbb{G}}_m\rightarrow X\times\mathbb{A}^1)_!(X\times{\mathbb{G}}_m\rightarrow X)^*\mathcal{F}[1]$. We deduce that ${\phi^{\operatorname{mon}}_{f}}\mathcal{F}=j_!j^*\mathcal{G}[1]$ where $\mathcal{G}=(X\times\mathbb{A}^1\xrightarrow{\pi} X)^*\mathcal{F}$ and $j\colon X\times{\mathbb{G}}_m\rightarrow X\times{\mathbb{A}}^1$ is the inclusion.  Let $i\colon X\times \{0\}\hookrightarrow X\times\mathbb{A}^1$ be the inclusion of the complement.  Since $\mathcal{G}$ is by definition trivial in $\operatorname{MMHM}(X)$, we deduce from the distinguished triangle $j_!j^*\rightarrow \operatorname{id}\rightarrow i_*i^*$ that ${\phi^{\operatorname{mon}}_{f}}\mathcal{F}\cong i_*\mathcal{F}$ in $\operatorname{MMHM}(X)$. In other words, ${\phi^{\operatorname{mon}}_{0}}\mathcal{F}\cong\mathcal{F}$, where we consider $\mathcal{F}$ on the right hand side of the isomorphism as a monodromic mixed Hodge module via pushforward along $X\xrightarrow{i} X\times{\mathbb{A}}^1$. It follows from exactness of $\phi_{f/u}[-1]$ (a result of Gabber [@BBD]), $\pi^*[1]$ and $j_!$ that ${\phi^{\operatorname{mon}}_{f}}$ is exact in general. 

We collect together some useful properties of ${\phi^{\operatorname{mon}}_{f}}$.

\[basicfacts\]

1.   \[monPD\](Poincaré duality): There is a natural isomorphism $${\mathbb{D}}^{\operatorname{mon}}_X{\phi^{\operatorname{mon}}_{f}}\cong {\phi^{\operatorname{mon}}_{f}} {\mathbb{D}}_X\colon {\mathcal{D}^{b}}(\operatorname{MHM}(X))\rightarrow{\mathcal{D}^{b}}(\operatorname{MMHM}(X)),$$ where ${\mathbb{D}}_X$ is the usual Verdier duality functor. 

2.   (Interaction with adjunction):\[adjInt\] If $p\colon X\rightarrow Y$ is a map of varieties, and $f$ is a regular function on $Y$, then there is a natural transformation ${\phi^{\operatorname{mon}}_{f}}p_*\rightarrow p_*{\phi^{\operatorname{mon}}_{fp}}$, which is moreover a natural isomorphism if $p$ is proper. 

3.   \[HomInv\](Homotopy invariance): Let $X'\rightarrow X$ be an affine fibration with $d$-dimensional fibres.  Then the natural map ${\phi^{\operatorname{mon}}_{f}} \mathcal{F}\rightarrow p_*{\phi^{\operatorname{mon}}_{f\circ p}}p^*\mathcal{F}$ is an isomorphism, as is the natural map $p_!{\phi^{\operatorname{mon}}_{f\circ p}}p^*\mathcal{F}\rightarrow {\phi^{\operatorname{mon}}_{f}}\mathcal{F}\otimes\operatorname{H}_c(\mathbb{A}^d)$.

4.   \[exFact\] (Exactness): The functor ${\phi^{\operatorname{mon}}_{f}}\colon {\mathcal{D}^{b}}(\operatorname{MHM}(X))\rightarrow{\mathcal{D}^{b}}(\operatorname{MMHM}(X))$ is exact, i.e. it restricts to an exact functor $\operatorname{MHM}(X)\rightarrow\operatorname{MMHM}(X)$.

5.   (Vanishing cycle decomposition theorem)\[vanDec\] Let $\mathcal{F}\in\operatorname{MMHM}(X)$ be pure, let $p\colon X\rightarrow Y$ be a proper map, and let $f$ be a regular function on $Y$. Then there is a non-canonical isomorphism $$p_*{\phi^{\operatorname{mon}}_{fp}}\mathcal{F}\cong \operatorname{\mathcal{H}}(p_*{\phi^{\operatorname{mon}}_{fp}}\mathcal{F}).$$

6.   (Thom–Sebastiani isomorphism): Let $f_j\colon X_j\rightarrow\mathbb{C}$ be regular functions, for $j=1,2$. Then there is a natural ismorphism of bifunctors $$\pi_1^*{\phi^{\operatorname{mon}}_{f_1}}(\bullet)\otimes\pi_2^*{\phi^{\operatorname{mon}}_{f_2}}(\bullet)\rightarrow{\phi^{\operatorname{mon}}_{f_1\boxplus f_2}}(\pi_1^*(\bullet)\otimes\pi_2^*(\bullet))|_{f^{-1}_1(0)\times f^{-1}_2(0)}.$$

7.   (Integral identity): Let $V_+\oplus V_-$ be a $\mathbb{C}^*$-equivariant vector bundle on the space $X$, given an $\mathbb{C}^*$-action, where the weights of the action on $V_+$ are all $1$, and the weights of the action on $V_-$ are all $-1$. Let $f$ be a $\mathbb{C}^*$-invariant function.  Below, for a vector bundle $V$ we write $T_{V}$ for the total space of $V$. Then the natural map $$(T_{V_+\oplus V_-}\rightarrow X)_!{\phi^{\operatorname{mon}}_{f}}(\mathbb{Q}_{T_{V_+\oplus V_-}}\rightarrow (T_{V_+}\rightarrow T_{V_+\oplus V_-})_*\mathbb{Q}_{T_{V_+}})$$ is an isomorphism. 

The first five statements follow from the corresponding statements for $\phi_f$. The first of these is proved at the level of perverse sheaves as the main theorem of [@Ma09], and for mixed Hodge modules as the main theorem of [@Sai89duality], see also Schürmann’s appendix to [@Br12].  The second statement is given by combining [@Saito90 Thm.2.14, Thm.4.3].  Homotopy invariance then follows from the homotopy invariance statement for perverse sheaves.  The hard part of the exactness statement is the statement that the shift of the usual vanishing cycle functor, $\phi_f[-1]$, is exact.  This is a result of Gabber, and can be found in [@BBD].  The version of the decomposition theorem quoted here follows from Saito’s version of the decomposition theorem [@Saito90], (\[exFact\]) and (\[adjInt\]).  The statement regarding the Thom–Sebastiani isomorphism is in fact false for $\phi_f$. The statement involving ${\phi^{\operatorname{mon}}_{f}}$ is due to Saito [@Saito10]; again we refer the reader to Schürmann’s appendix to [@Br12] for the compatibility between this proof and the proof of Massey [@Ma01] at the level of complexes of perverse sheaves.  The integral identity is proved in the above form in [@KS2].  In fact it is the only property that we will not use in this paper. 

For ease of exposition we make the following simplification in what follows. 

\[phicheat\] For all functions $f\colon X\rightarrow \mathbb{C}$ for which we wish to take ${\phi^{\operatorname{mon}}_{f}}$, if $X$ is smooth, we assume that $\operatorname{crit}(f)\subset f^{-1}(0)$ as sets. 

Under the assumption, the Thom–Sebastiani isomorphism for constant mixed Hodge modules simplifies to $$\pi_1^*{\phi^{\operatorname{mon}}_{f_1}}({\mathbb{Q}}_{X_1})\otimes\pi_2^*{\phi^{\operatorname{mon}}_{f_2}}({\mathbb{Q}}_{X_1})\rightarrow{\phi^{\operatorname{mon}}_{f_1\boxplus f_2}}({\mathbb{Q}}_{X_1\times X_2}).$$ The assumption can be dropped, with a little care.  If it does not hold, one should instead work with the functor $\phi^{\operatorname{mon},\textrm{fib}}_f:=\bigoplus_{a\in\mathbb{A}_{\mathbb{C}}^1}{\phi^{\operatorname{mon}}_{f-a}}$ – see [@DaMe4]. 

Given an element $\mathcal{G}\in\operatorname{MMHM}(X)$, we say it is pure of weight $i$ if $\operatorname{Gr}_W^j\mathcal{G}$ is zero for all $j\neq i$. Given an element $\mathcal{F}\in{\mathcal{D}}(\operatorname{MMHM}(X))$, we say that $\mathcal{F}$ is pure of weight $i$ if each $\operatorname{\mathcal{H}}^j(\mathcal{F})$ is pure of weight $i+j$, or we just say $\mathcal{F}$ is *pure* if it is pure of weight zero. 

Consider the embedding ${\mathcal{D}^{b}}(\operatorname{MHM}(\operatorname{pt}))\xrightarrow{i_*}{\mathcal{D}^{b}}(\operatorname{MMHM}(\operatorname{pt}))$. The former category contains the element $$\mathbb{L}:=\operatorname{H}_c(\mathbb{A}^1,\mathbb{Q}),$$ which is pure.  There is no square root of $\mathbb{L}$ in ${\mathcal{D}^{b}}(\operatorname{MHM}(\operatorname{pt}))$, i.e. an element $\mathbb{L}^{1/2}$ such that $(\mathbb{L}^{1/2})^{\otimes 2}\cong \mathbb{L}$, but after embedding in ${\mathcal{D}^{b}}(\operatorname{MMHM}(\operatorname{pt}))$ there is a choice of square roots.  We set $\mathbb{L}^{1/2}:=\operatorname{H}(\mathbb{A}^1,{\phi^{\operatorname{mon}}_{x^2}}\mathbb{Q}_{\mathbb{A}^1})$, where $x^2\colon \mathbb{A}^1\rightarrow\mathbb{C}$ is considered as a regular function, and $\mathbb{Q}_{\mathbb{A}^1}$ is the constant mixed Hodge module on $\mathbb{A}^1$. Since ${\mathbb{L}}^{1/2}$ is the tensor square root of $\mathbb{L}$, we deduce that it is pure, concentrated in cohomological degree 1, by Remark \[convProdDef\].  Note that due to the definition of the category of mixed Hodge modules, this cohomological degree is with respect to the perverse t structure on the underlying category of constructible sheaves on $\mathbb{A}^1$.

If $\mathcal{F}\in\operatorname{MMHM}(X)$ and $\mathcal{G}\in\operatorname{MMHM}(\operatorname{pt})$ we use the abbreviation $$\mathcal{F}\otimes\mathcal{G}:=\mathcal{F}\otimes 
\tau^*\mathcal{G},$$ where $\tau\colon X\rightarrow \operatorname{pt}$ is the map to a point.  If $X$ is a smooth equidimensional variety, we define ${\mathcal{IC}}_{X}(\mathbb{Q}):=\mathbb{Q}_X\otimes{\mathbb{L}}^{-\dim(X)/2}$. This is pure, and an element of $\operatorname{MMHM}(X)$, again by Remark \[convProdDef\].  If $X$ is not smooth, but is irreducible, we define ${\mathcal{IC}}_X(\mathbb{Q}):={\tilde{\mathcal{IC}}}_X(\mathbb{Q}_{X_{\operatorname{reg}}})\otimes{\mathbb{L}}^{-\dim(X_{\operatorname{reg}})/2}$, where ${\tilde{\mathcal{IC}}}_X(\mathbb{Q}_{X_{\operatorname{reg}}})$ is the usual intersection cohomology mixed Hodge module complex of $X$. If $X$ is a disjoint union of irreducible varieties, we define $${\mathcal{IC}}_X(\mathbb{Q}):=\bigoplus_{Z\in\pi_0(X)}{\mathcal{IC}}_Z(\mathbb{Q}),$$ and define $${\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_X(\mathbb{Q})=\bigoplus_{Z\in\pi_0(X)}\left({\phi^{\operatorname{mon}}_{f}}{\tilde{\mathcal{IC}}}_Z(\mathbb{Q}_{Z_{\operatorname{reg}}})\otimes{\mathbb{L}}^{-\dim(Z_{\operatorname{reg}})/2}\right).$$

Because of the shift in the definition of ${\mathcal{IC}}_X(\mathbb{Q})$, there are natural isomorphisms ${\mathbb{D}}^{\operatorname{mon}}_X{\mathcal{IC}}_X(\mathbb{Q})\cong{\mathcal{IC}}_X(\mathbb{Q})$ and ${\mathbb{D}}^{\operatorname{mon}}_X{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_X(\mathbb{Q})\cong{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_X(\mathbb{Q})$ for all $X$.

For $X$ a smooth equidimensional variety we define $$\begin{aligned}
\operatorname{H}(X,\mathbb{Q})_{\operatorname{vir}}:=&\operatorname{H}(X,{\mathcal{IC}}_X(\mathbb{Q}))\\\cong&\operatorname{H}(X,\mathbb{Q})\otimes{\mathbb{L}}^{-\dim(X)/2}\in{\mathcal{D}^{b}}(\operatorname{MMHM}(\operatorname{pt})),\end{aligned}$$ and observing that $\mathbb{C}\mathbb{P}^{\infty}\cong \textrm{B}\mathbb{C}^*$, which has dimension $-1$ as a stack, extend this notation by defining $$\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty},\mathbb{Q})_{\operatorname{vir}}:=\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty},\mathbb{Q})\otimes{\mathbb{L}}^{1/2}\in {\mathcal{D}}(\operatorname{MMHM}(\operatorname{pt})).$$

Equivariant vanishing cycles {#TotCon}
----------------------------

We do not propose here to write down a full six functor and vanishing cycles functor formalism for equivariant monodromic mixed Hodge modules via a theory of monodromic mixed Hodge modules on stacks, instead mimicking the constructions of [@BL94] and Totaro’s approximation [@To99] of the Chow ring of classifying spaces by finite-dimensional algebraic approximations to produce only the definitions and constructions we will need for the rest of this paper. 

We will want to be able to work with equivariant monodromic mixed Hodge modules, in the following generality.  We assume that we have a $G$-action on a smooth algebraic variety $X$, for $G$ an affine algebraic group, and a regular function $f$ on the stack $X/G$, i.e. a $G$-invariant regular function on $X$. Furthermore we assume that we are given a map of stacks $p\colon X/G\rightarrow Y$, where $Y$ is another complex variety.  We will also assume that $G$ is special, i.e. all étale locally trivial principal $G$-bundles are Zariski locally trivial – see [@JoyceMF Def.2.1] for a concise discussion of this condition in this context, or [@Chev58] for the original references.  As noted in [@JoyceMF], all such $G$ are connected. 

The element $\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_X(\mathbb{Q})\right)$ will be an element of ${\mathcal{D}^{\geq}}(\operatorname{MMHM}(Y))$, the derived category of bounded below complexes of monodromic mixed Hodge modules on $Y$.

We define $\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_X(\mathbb{Q})\right)$ as follows.  Let $V_1\subset V_2\subset\ldots$ be an ascending chain of $G$-representations, and let $U_1\subset U_2\subset\ldots$ be an ascending chain of subvarieties of the underlying vector spaces of $V_1,V_2,\ldots$, considered as $G$-equivariant algebraic varieties.  We suppose that $G$ acts scheme-theoretically freely on each $U_i$, that each principal bundle quotient $U_i\rightarrow U_i/G$ exists in the category of schemes, and that $\operatorname{codim}_{V_i}(V_i\setminus U_i)\rightarrow \infty$ as $i\rightarrow\infty$. The map $X\times U_i\rightarrow X\times_G U_i$ exists as a principal bundle quotient in the category of schemes by [@EdGr98 Prop.23].  We define $f_i\colon X\times_G U_i\rightarrow \mathbb{C}$ to be the induced map, and $\iota_i\colon X\times_G U_i\rightarrow X\times_G U_{i+1}$ to be the inclusion. 

To fix notation, we fix an embedding $G\subset \operatorname{GL}_t(\mathbb{C})$ for some $t$. We set $$V_i:=\operatorname{Hom}(\mathbb{C}^i,\mathbb{C}^t),$$ and define $U_i\subset V_i$ to be the subscheme of surjective maps – if $i\geq t$ then $U_i$ does indeed carry a free $G$-action via the $\operatorname{GL}_t(\mathbb{C})$-action on $\mathbb{C}^t$. Arguing as in [@EdGr98 Prop.23] there is a covering of $X$ by quasiprojective subvarieties $Y_i$ carrying $\operatorname{GL}_t(\mathbb{C})$-equivariant line bundles, relatively ample for the projection $Y_i\times U\rightarrow U$, so by the induced $G$-invariance and [@MFK94 Prop.7.1] we deduce that there is a principal $G$-bundle quotient $X\times U\rightarrow X\times_G U$. For each $i$, we define $X_i:=X\times_G U_i$. We obtain an explicit sequence of maps $$p_{i+1,*}{\phi^{\operatorname{mon}}_{f_i+1}}\mathbb{Q}_{X_{i+1}}\rightarrow p_{i+1,*}\iota_{i,*}{\phi^{\operatorname{mon}}_{f_i}}\mathbb{Q}_{X_i}= p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}\mathbb{Q}_{X_i}$$ and so a sequence of mixed Hodge modules $$\mathcal{F}_i:=p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}{\mathbb{Q}}_{X_i}.$$

\[stabProp\] Fix $n\in\mathbb{N}$. Then for $i\gg 0$ the map $$\label{evIs}
\Lambda\colon \mathcal{H}^{n}(Y,\mathcal{F}_{i+1})\rightarrow \mathcal{H}^n(Y,\mathcal{F}_i)$$ in $\operatorname{MMHM}(Y)$ is an isomorphism. 

Since the functor ${\operatorname{forg}^{\operatorname{mon}}_{X}}\colon \operatorname{MMHM}(Y)\rightarrow\operatorname{Perv}(Y)$ of equation (\[formDef\]) is faithful, it suffices to show that (\[evIs\]) induces an isomorphism at the level of perverse sheaves, with $\mathcal{F}_i$ replaced by $p_{i,*}\phi_{f_i}\mathbb{Q}_{X_i}$. Say we can prove the same proposition, but with the map $\Lambda$ replaced by the map $$\Lambda_{\operatorname{con}}\colon \mathcal{H}_{\operatorname{con}}^{n}(Y,\mathcal{F}_{i+1})\rightarrow \mathcal{H}_{\operatorname{con}}^n(Y,\mathcal{F}_i)$$ of constructible sheaves on $Y$ (here and below we use $\operatorname{\mathcal{H}}_{\operatorname{con}}$ to denote constructible cohomology sheaves).  Then for sufficiently large $i$, $$\operatorname{cone}(\mathcal{F}_{i+1}\rightarrow\mathcal{F}_i)\in \mathcal{D}_{\operatorname{con}}^{\geq n+1}(Y)\subset {{}^{\mathfrak{p}}\mathcal{D}}^{\geq n+1}(Y)$$ and the proposition follows.  Consider the space $U_{i,i+1}\subset \operatorname{Hom}(\mathbb{C}^{i+1},\mathbb{C}^t)$ of linear maps which are surjective after precomposing with the projection $$\pi_{i,i+1}\colon \mathbb{C}^{i+1}\rightarrow \mathbb{C}^{i}$$ given by $(z_1,\ldots,z_{i+1})\mapsto (z_1,\ldots,z_i)$. We denote by $j_i$ the inclusion $$j_i\colon U_{i,i+1}\hookrightarrow U_{i+1}.$$ Note that for large $i$, $G$ acts freely on $U_{i,i+1}$, and there is a $G$-equivariant affine fibration $\tau_{i,i+1}\colon U_{i,i+1}\rightarrow U_i$ given by $\tau_{i,i+1}f(z):= f(z,0)$.

Define $X_{i,i+1}:=X\times_G U_{i,i+1}$, denote by $f_{i,i+1}\colon  X_{i,i+1}\rightarrow \mathbb{C}$ the function induced by $f$, and denote by $$\label{taffDef}
\iota_{i,i+1}\colon X_{i,i+1}\rightarrow X_{i+1}$$ the open embedding induced by $j_{i}$. The projection $\tau_{i,i+1}$ induces an affine fibration $t_{i,i+1}\colon X_{i,i+1}\rightarrow X_{i}$ with zero section $$z_i\colon X_i\rightarrow X_{i,i+1},$$ and we define the induced function $$p_{i,i+1}\colon X_{i,i+1}\xrightarrow{t_{i,i+1}}X_i\xrightarrow{p_i} Y.$$ We factorise $\Lambda_{\operatorname{con}}$ as the composition $$\operatorname{\mathcal{H}}^n_{\operatorname{con}}(p_{i+1,*}\phi_{f_{i+1}}\mathbb{Q}_{X_{i+1}})\xrightarrow{a} \operatorname{\mathcal{H}}^n_{\operatorname{con}}(p_{i+1,*}\iota_{i,i+1,*}\phi_{f_{i,i+1}}\mathbb{Q}_{X_{i,i+1}})\rightarrow \operatorname{\mathcal{H}}(p_{i,*}\phi_{f_i}\mathbb{Q}_{X_i})$$ where the second map is an isomorphism by homotopy invariance.  So it is enough to show that the map $a$ is an isomorphism for sufficiently large $i$.

Consider the function $\overline{f}_{i+1}\colon X\times U_{i+1}\rightarrow \mathbb{C}$ given by the composition $X\times U_{i+1}\xrightarrow{\pi } X\xrightarrow{f}\mathbb{C}$, and define $\overline{f}_{i,i+1}\colon X\times U_{i,i+1}\rightarrow \mathbb{C}$ similarly.  Let $\overline{\iota}_{i,i+1}\colon  X\times U_{i,i+1}\rightarrow X\times U_{i+1}$ be the inclusion.  Then $$\phi_{\overline{f}_{i+1}}\mathbb{Q}_{X\times U_{i+1}}\cong\phi_{f}\mathbb{Q}_{X}\boxtimes \mathbb{Q}_{U_{i+1}}$$ and $$\left(\phi_{\overline{f}_{i+1}}\mathbb{Q}_{X\times U_{i+1}}\rightarrow\overline{\iota}_{i,i+1,*}\phi_{\overline{f}_{i,i+1}}\mathbb{Q}_{X\times U_{i,i+1}}\right)=\operatorname{id}_{\phi_{f}\mathbb{Q}_{X}}\boxtimes (\mathbb{Q}_{U_{i+1}}\rightarrow j_{i,*}\mathbb{Q}_{U_{i,i+1}}).$$ For fixed $m$, $\operatorname{\mathcal{H}}_{\operatorname{con}}^m(\mathbb{Q}_{U_{i+1}})\rightarrow \operatorname{\mathcal{H}}_{\operatorname{con}}^m(\mathbb{Q}_{U_{i,i+1}})$ is an isomorphism for sufficiently large $i$, since the codimension of $U_{i+1}\setminus U_{i,i+1}$ inside $U_{i+1}$ goes to infinity as $i$ goes to infinity.  Since the external tensor product is exact, we deduce that for fixed $m$ and sufficiently large $i$, $$\operatorname{\mathcal{H}}^m_{\operatorname{con}}(\phi_{\overline{f}_{i+1}}\mathbb{Q}_{X\times U_{i+1}}\rightarrow\overline{\iota}_{i,i+1,*}\phi_{\overline{f}_{i,i+1}}\mathbb{Q}_{X\times U_{i,i+1}})$$ is an isomorphism.  On the other hand, taking a Zariski open subspace $C\subset X_{i+1}$ such that the principal bundle $\overline{C}=C\times_{X_{i+1}}(X\times U_{i+1})\rightarrow C$ is trivial, we have that $$\begin{aligned}
&\operatorname{\mathcal{H}}_{\operatorname{con}}(\phi_{\overline{f}_{i+1}}\mathbb{Q}_{X\times U_{i+1}}\rightarrow\overline{\iota}_{i,i+1,*}\phi_{\overline{f}_{i,i+1}}\mathbb{Q}_{X\times U_{i,i+1}})|_{\overline{C}}\cong\\& \operatorname{\mathcal{H}}_{\operatorname{con}}(\phi_{f_{i+1}}\mathbb{Q}_{X_{i+1}}\rightarrow\iota_{i,i+1,*}\phi_{f_{i,i+1}}\mathbb{Q}_{X\times_{G} U_{i,i+1}}))\boxtimes\operatorname{id}_{\mathbb{Q}_{G}}.    \end{aligned}$$ and so using exactness of external tensor product again, we deduce that for fixed $n$ and sufficiently large $i$, the map $$\operatorname{\mathcal{H}}^n_{\operatorname{con}}(\phi_{f_{i+1}}\mathbb{Q}_{X_{i+1}}\rightarrow\iota_{i,i+1,*}\phi_{f_{i,i+1}}\mathbb{Q}_{X_{i,i+1}}))$$ is an isomorphism.  The pushforward $p_{i+1,*}$ maps $\mathcal{D}_{\operatorname{con}}^{\geq n+1}(X_{i+1})\rightarrow \mathcal{D}_{\operatorname{con}}^{\geq n+1}(Y)$, and so $$\operatorname{cone}\left(p_{i+1,*}\phi_{f_{i+1}}\mathbb{Q}_{X_{i+1}}\rightarrow p_{i+1,*}\iota_{i,i+1,*}\phi_{f_{i,i+1}}\mathbb{Q}_{X_{i,i+1}}\right)\in\mathcal{D}_{\operatorname{con}}^{\geq n+1}(X)$$ and the map $a$ is an isomorphism, as required. 

We define $\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_X\right)$ to be the complex $$\ldots\xrightarrow{0}\operatorname{\mathcal{H}}^{n-1}(\lim_{i\mapsto\infty}p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}\mathbb{Q}_{X_i})\xrightarrow{0}\operatorname{\mathcal{H}}^{n}(\lim_{i\mapsto\infty}p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}\mathbb{Q}_{X_i})\xrightarrow{0}\ldots$$ which is well defined by Proposition \[stabProp\].  If ${\mathfrak{M}}=\coprod_{s\in S} X_s/G_s$, with each $X_s$ a smooth algebraic variety, and each $G_s$ a special affine algebraic group, and $p\colon {\mathfrak{M}}\rightarrow Y$ is a map to an algebraic variety, we define $$\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{{\mathfrak{M}}}\right):=\bigoplus_{s\in S} \operatorname{\mathcal{H}}(p_*{\phi^{\operatorname{mon}}_{f|_{X_s/G_s}}}\mathbb{Q}_{X_s/G_s})$$ and $$\operatorname{\mathcal{H}}(p_*{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_{{\mathfrak{M}}}(\mathbb{Q})):=\bigoplus_{s\in S}\operatorname{\mathcal{H}}(p_*{\phi^{\operatorname{mon}}_{f|_{X_s/G_s}}}\mathbb{Q}_{X_s/G_s})\otimes \mathbb{L}^{(\dim(G_s)-\dim(X_s))/2}.$$

We define $\operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{{\mathfrak{M}}}\right)\in {\mathcal{D}^{\leq}}(\operatorname{MMHM}(Y))$ in similar fashion.  We give the definition in the case that ${\mathfrak{M}}=X/G$, for $X$ a smooth irreducible variety and $G$ a special affine algebraic group, and extend to the case of a disjoint union of such stacks as above.  Consider the isomorphism $\operatorname{\mathcal{H}}(\iota_{i,i+1,!}{\phi^{\operatorname{mon}}_{f_{i,i+1}}}\mathbb{Q}_{X_{i,i+1}})\cong \operatorname{\mathcal{H}}({\phi^{\operatorname{mon}}_{f_{i}}}\mathbb{Q}_{X_{i}})\otimes\mathbb{L}^{\dim(X_{i+1})-\dim(X_i)}$ where $\iota_{i,i+1}$ is as defined in (\[taffDef\]).  Arguing as in Proposition \[stabProp\], for fixed $n$ the natural map $$\operatorname{\mathcal{H}}_{\operatorname{con}}^{2\dim(U_{i+1})+n}((X_{i,i+1}\rightarrow X_{i+1})_!{\phi^{\operatorname{mon}}_{f_{i,i+1}}}\mathbb{Q}_{X_{i,i+1}})\rightarrow \operatorname{\mathcal{H}}_{\operatorname{con}}^{2\dim(U_{i+1})+n}({\phi^{\operatorname{mon}}_{f_{i+1}}}\mathbb{Q}_{X_{i+1}})$$ is an isomorphism for sufficiently large $i$, and we can define $\operatorname{\mathcal{H}}^n\left(p_!{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)$ to be the limit of $\operatorname{\mathcal{H}}^n\left(p_{i,!}{\phi^{\operatorname{mon}}_{f_i}}\mathbb{Q}_{X_i}\otimes \mathbb{L}^{-\dim(U_i)}\right)$ as $i$ tends to infinity, and $$\operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_{X/G}(\mathbb{Q})\right):=\operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)\otimes{\mathbb{L}}^{(\dim(G)-\dim(X))/2}.$$

Note that we do not offer here a definition of $p_*\phi_{\operatorname{mon},f}\mathbb{Q}_{X/G}$, or $p_!\phi_{\operatorname{mon},f}\mathbb{Q}_{X/G}$ as objects of the derived category, but instead limit ourselves to defining the total cohomology of these direct images with respect to the natural t structure on ${\mathcal{D}}(\operatorname{MMHM}(Y))$ (i.e. the pullback of the natural t structure on ${\mathcal{D}}(\operatorname{Perv}(X))$ under ${\operatorname{forg}^{\operatorname{mon}}_{X}}$).  Of course in case $X/G$ is an actual scheme, $p_*\phi^{\operatorname{mon}}_{f}\mathbb{Q}_{X/G}$ and $p_!\phi^{\operatorname{mon}}_{f}\mathbb{Q}_{X/G}$ are well defined before passing to cohomology, and our definition recovers their respective total perverse cohomology. 

As a special case of the above construction, if a map $\tau\colon X/G\rightarrow \mathbb{N}^{Q_0}$ to a semigroup of dimension vectors is understood, we define $$\begin{aligned}
\operatorname{H}(X/G,{\phi^{\operatorname{mon}}_{f}}{\mathbb{Q}}_{X/G}):=\operatorname{\mathcal{H}}(\tau_*{\phi^{\operatorname{mon}}_{f}}{\mathbb{Q}}_{X/G}).\end{aligned}$$ This is the definition used in [@KS2 Sec.7] in the definition of the underlying $\mathbb{N}^{Q_0}$-graded monodromic mixed Hodge module of the critical cohomological Hall algebra. 

Let $h\colon X'\rightarrow X$ be a morphism of $G$-equivariant varieties.  Then for each $i$ we obtain maps $h_i\colon X'_i\rightarrow X_i$. There are natural maps $$p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}h_{i,*}\mathbb{Q}_{X'_i}\rightarrow p_{i,*}h_{i,*}{\phi^{\operatorname{mon}}_{f_i\circ h_i}}\mathbb{Q}_{X'_i}$$ which are isomorphisms if $h$ is proper or an affine fibration (see Remark \[basicfacts\]).  We precompose with the natural map $$p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}\mathbb{Q}_{X_i}\rightarrow p_{i,*}{\phi^{\operatorname{mon}}_{f_i}}h_{i,*}\mathbb{Q}_{X'_i}$$ and let $i\mapsto \infty$, defining maps $$\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)\rightarrow \operatorname{\mathcal{H}}\left((p h)_*{\phi^{\operatorname{mon}}_{f\circ h}}\mathbb{Q}_{X'/G}\right).$$ We define the map $$\operatorname{\mathcal{H}}\left((p h)_!){\phi^{\operatorname{mon}}_{f\circ h}}\mathbb{Q}_{X'/G}\right)\rightarrow\operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)$$ in the same way, starting from the maps $ p_{i,!}{\phi^{\operatorname{mon}}_{f_i}}h_{i,!}{\mathbb{D}}_{X'_i}\mathbb{Q}_{X'_i}\rightarrow p_{i,!}{\phi^{\operatorname{mon}}_{f_i}}{\mathbb{D}}_{X_i}\mathbb{Q}_{X_i}$ . 

Finally, let $\upsilon\colon H\rightarrow G$ be an inclusion of groups – the only examples we will consider are when $\upsilon$ is the inclusion of a parabolic subgroup inside $\operatorname{GL}_n(\mathbb{C})$, or the inclusion $L\subset P$ of the Levi subgroup of a parabolic subgroup of $\operatorname{GL}_n(\mathbb{C})$. Let $G$ act on $X$ as above, with $f$ a $G$-invariant regular function on $X$. Let $h\colon X/H\rightarrow X/G$ be the associated morphism of stacks.  Then we obtain maps $$\begin{aligned}
X\times_H U_i\xrightarrow{h_i} &X\times_G U_i\\
(x,z)\mapsto&(x,z)\end{aligned}$$ which we use in the same way as above to obtain the map $$\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)\rightarrow\operatorname{\mathcal{H}}\left((p h)_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/H}\right)$$ and the map $$\operatorname{\mathcal{H}}\left((p h)_!{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/P}\right)\rightarrow \operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)\otimes{\mathbb{L}}^{\dim(G)-\dim(P)}.$$

\[limKer\] If $H\hookrightarrow G$ is the inclusion of a parabolic subgroup, then the induced maps $h_i$ are proper.  As such, there is a natural isomorphism ${\phi^{\operatorname{mon}}_{f_i}}h_{i,*}{\mathbb{Q}}_{X\times_H U_i}\cong h_{i,*}{\phi^{\operatorname{mon}}_{f_i}}{\mathbb{Q}}_{X\times_H U_i}$. From the maps $h_{i,*}{\mathbb{D}}_{X\times_H U_i}\mathbb{Q}_{X\times_H U_i}\rightarrow {\mathbb{D}}_{X\times_G U_{i}}\mathbb{Q}_{X\times_G U_{i}}$ given by the natural isomorphism $h_{i,*}\cong h_{i,!}$ and Verdier duality, we obtain, in the limit, the map $$\operatorname{\mathcal{H}}\left((ph)_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/H}\right)\rightarrow \operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{f}}\mathbb{Q}_{X/G}\right)\otimes{\mathbb{L}}^{\dim(G)-\dim(H)}.$$

Moduli spaces of quiver representations {#Qrepsection}
=======================================

Basic notions
-------------

We use the notations and conventions from [@DaMe4], which we briefly recall.  Let $Q=(Q_0,Q_1,s,t)$ denote a quiver, that is, a pair of finite sets $Q_0$ and $Q_1$, and a pair of maps $s\colon Q_1\rightarrow Q_0$ and $t\colon Q_1\rightarrow Q_0$, the maps taking an arrow to its source and target, respectively.  Denote by $\mathbb{C} Q$ the free path category of $Q$ over $\mathbb{C}$. Alternatively we may think of $\mathbb{C} Q$ as the free path algebra of $Q$, with a distinguished family of mutually orthogonal idempotents $e_i$ in bijection with the vertices $Q_0$, summing to $1_{\mathbb{C} Q}$.

Let $\operatorname{Sch}_{\mathbb{C}}$ be the category of schemes over $\operatorname{Spec}(\mathbb{C})$. For $S\in\operatorname{Sch}_{\mathbb{C}}$ we denote by $\operatorname{Vect}_{S}^{\operatorname{f.d}}$ the category of finite rank vector bundles over $S$. Let $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$ be a dimension vector.  We denote by ${\mathfrak{M}}_{\operatorname{\mathbf{d}}}$ the groupoid valued functor on $\operatorname{Sch}_{\mathbb{C}}$ defined by setting ${\mathfrak{M}}_{\operatorname{\mathbf{d}}}(S)$ to be the groupoid obtained from forgetting the non-invertible morphisms in the category of functors in $\operatorname{Fun}(\mathbb{C} Q,\operatorname{Vect}_{S}^{\operatorname{f.d}})$ such that $i\in Q_0$ is sent to a vector bundle of dimension $\operatorname{\mathbf{d}}_i$. This prestack is an Artin stack, as it is represented by the following global quotient stack.  First define $$\label{Xdef}
X_{\operatorname{\mathbf{d}}}:=\prod_{a\in Q_1}\operatorname{Hom}(\mathbb{C}^{\operatorname{\mathbf{d}}_{s(a)}},\mathbb{C}^{\operatorname{\mathbf{d}}_{t(a)}}).$$ This affine space carries the change of basis action of $$G_{\operatorname{\mathbf{d}}}:=\prod_{i\in Q_0}\operatorname{Aut}(\mathbb{C}^{\operatorname{\mathbf{d}}_i}),$$ and there is an equivalence of stacks ${\mathfrak{M}}_{\operatorname{\mathbf{d}}}\cong X_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}$. We denote by ${\mathfrak{M}}$ the union $\coprod_{\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}} {\mathfrak{M}}_{\operatorname{\mathbf{d}}}$, the stack of finite-dimensional representations of $Q$, which by the equivalences just given is a countable disjoint union of finite type global quotient Artin stacks.  For the rest of the paper, $X_{\operatorname{\mathbf{d}}}$ will be as in (\[Xdef\]).  Where we wish to be specific regarding the quiver with respect to which $X_{\operatorname{\mathbf{d}}}$ is defined, we will instead use the notation $X(Q)_{\operatorname{\mathbf{d}}}$. For $\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''=\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$ let ${\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}(S)$ be the groupoid of triples $(F',F,\iota)$, where $F',F\in\operatorname{Fun}(\mathbb{C}Q,\operatorname{Vect}_S^{\operatorname{f.d}})$, and $\iota\colon F'\rightarrow F$ is a natural transformation such that $\dim(F'(i))=\operatorname{\mathbf{d}}'_i$, $\dim(F(i))=\operatorname{\mathbf{d}}_i$, and $\iota(i)$ is injective, with locally free cokernel, for every $i$. Again, ${\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ is a finite type Artin stack, which can be described as follows.  Let $X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\subset X_{\operatorname{\mathbf{d}}}$ be the subspace of representations such that the flag $\mathbb{C}^{\operatorname{\mathbf{d}}'_i}\subset \mathbb{C}^{\operatorname{\mathbf{d}}_i}$ is preserved for all $i\in Q_0$, and let $G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\subset G_{\operatorname{\mathbf{d}}}$ be the subgroup preserving these same flags.  Then $${\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\cong X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}.$$

A tuple $\zeta=(\zeta_i)_{i\in Q_0}\in \mathbb{H}_+^{Q_0}:=\{r\exp(i\pi\phi)\in \mathbb{C}\mid r>0, 0<\phi\le 1\}^{Q_0}\subset \mathbb{C}^{Q_0}$ provides a Bridgeland stability condition, as defined in [@Bridgeland02], with *central charge* defined on finite-dimensional $\mathbb{C}Q$-modules $$Z\colon\rho\mapsto\zeta\cdot \dim (\rho)=\sum_{i\in Q_0}\zeta_i\dim (\rho_i).$$ We define the *slope* of a representation $\rho$ by setting $${\Xi}^{\zeta}(\rho):=\begin{cases}- \Re e (Z(\rho))/ \Im m (Z(\rho))&\textrm{if }\Im m (Z(\rho))\neq 0\\\infty&\textrm{if }\Im m (Z(\rho))=0.\end{cases}$$ Likewise we define ${\Xi}^{\zeta}(\operatorname{\mathbf{d}})$, for $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$, to be the slope of any representation $\rho$ of dimension $\operatorname{\mathbf{d}}$. A $\mathbb{C} Q$-module $\rho$ is called $\zeta$*-semistable* if for all proper submodules $\rho'\subset\rho$ we have ${\Xi}^{\zeta}(\rho')\leq{\Xi}^{\zeta}(\rho)$, and is $\zeta$*-stable* if instead we have ${\Xi}^{\zeta}(\rho')<{\Xi}^{\zeta}(\rho)$ for every proper submodule.  To simplify notation in what follows, we will assume that ${\Xi}^{\zeta}(\rho)<\infty$ in this paper.  This is equivalent to the condition that $\zeta$ maps nonzero elements of $\mathbb{N}^{Q_0}$ to the open upper half plane.  We define two pairings on $\mathbb{Z}^{Q_0}$: $$(\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}):=\sum_{i\in Q_0} \operatorname{\mathbf{d}}_i \operatorname{\mathbf{e}}_i-\sum_{a\in Q_1}\operatorname{\mathbf{d}}_{s(a)}\operatorname{\mathbf{e}}_{t(a)}$$ and $$\langle \operatorname{\mathbf{d}},\operatorname{\mathbf{e}}\rangle:=(\operatorname{\mathbf{d}},\operatorname{\mathbf{e}})-(\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}).$$ Note that $(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})=-\dim {\mathfrak{M}}_{\operatorname{\mathbf{d}}}$. As in the introduction, for every $\mu\in (-\infty,\infty)$ we denote by $\Lambda_{\mu}^{\zeta}\subset\mathbb{N}^{Q_0}$ the submonoid of dimension vectors $\operatorname{\mathbf{d}}$ of slope $\mu$.

We say $\zeta$ is $\mu$-generic if $\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}\in\Lambda_{\mu}^{\zeta}$ implies $\langle \operatorname{\mathbf{d}},\operatorname{\mathbf{e}}\rangle=0$. We say $\zeta$ is generic if it is $\mu$-generic for all $\mu$.

We say that $\zeta$ is a King stability condition if $\Im m(\zeta_i)=1$ and $\Re e(\zeta_i)\in\mathbb{Q}$ for all $i\in Q_0$. Given a King stability condition $\zeta$, we can fix $m\in\mathbb{N}$ such that $m\Re e(\zeta_i)\in\mathbb{Z}$ for every $i$. We linearize the $G_{\operatorname{\mathbf{d}}}$-action on $X_{\operatorname{\mathbf{d}}}$ via the character $$\begin{aligned}
\label{charDef}
\chi_{\operatorname{\mathbf{d}}}\colon &G_{\operatorname{\mathbf{d}}}\rightarrow \mathbb{C}^*\\&(g_i)_{i\in Q_0}\mapsto \prod_{i\in Q_0}\det(g_i)^{m\Re e(\zeta_i)}, \nonumber\end{aligned}$$ and define $X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$ to be the variety of semistable points with respect to this linearization.  By [@King], using the constructions and definitions of [@MFK94], the GIT quotient $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}/\!\!/_{\chi_{\operatorname{\mathbf{d}}}} G_{\operatorname{\mathbf{d}}}$ provides a coarse moduli space of $\zeta$-semistable representations of dimension ${\operatorname{\mathbf{d}}}$, which we denote ${\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$.

\[BtoK\] Fix a dimension vector $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$ of slope $\mu$, for a $\mu$-generic stability condition $\zeta$. Then we can always find a $\mu'$-generic King stability condition $\zeta'$ such that a $\operatorname{\mathbf{d}}$-dimensional $\mathbb{C} Q$-module is $\zeta$-stable if and only if it is $\zeta'$-stable, by [@DMSS15 Lem.4.21], where $\mu'$ is the slope of $\operatorname{\mathbf{d}}$ with respect to $\zeta'$. By construction, a $\operatorname{\mathbf{d}}$-dimensional $\mathbb{C}Q$-module is $\zeta'$-semistable if and only if it is $\zeta$-semistable.  We deduce that for every $\mu$-generic Bridgeland stability condition $\zeta$ and every dimension vector $\operatorname{\mathbf{d}}$ of slope $\mu$, there is a coarse moduli space ${\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$ of $\operatorname{\mathbf{d}}$-dimensional $\zeta$-semistable $\mathbb{C}Q$-modules. 

For a slope $\mu$, we define $${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}=\coprod_{{\Xi}^{\zeta}(\operatorname{\mathbf{d}})=\mu}{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$$ and $${\mathfrak{M}}_{\mu}^{\zeta\operatorname{-ss}}=\coprod_{\Xi^{\zeta}(\operatorname{\mathbf{d}})=\mu}{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}.$$ We denote by $$p_{\operatorname{\mathbf{d}}}^{\zeta}\colon {\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\rightarrow {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$$ and $$p_{\mu}^{\zeta}\colon {\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}\rightarrow {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}$$ the maps from the stacks to their respective coarse moduli spaces, and by $$\label{qdef}
q_{\operatorname{\mathbf{d}}}^{\zeta}\colon {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\rightarrow {\mathcal{M}}_{\operatorname{\mathbf{d}}}$$ and $$\label{qmudef}
q_{\mu}^{\zeta}\colon {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}\rightarrow {\mathcal{M}}_{\mu}$$ the maps to the affinizations. 

Pick a slope $\mu\in (-\infty,\infty)$. Then we can define a maximally degenerate stability condition, for which every $\mathbb{C}Q$-module is automatically semistable, by fixing each $\zeta_i$ to be an element satisfying $- \Re e(\zeta_i) /  \Im m (\zeta_i)=\mu$. In this case we have $\Lambda_{\mu}^{\zeta}=\mathbb{N}^{Q_0}$, and ${\mathfrak{M}}={\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}$. As a result, all the results in this paper in which we do not assume that we are working with a generic stability condition apply to the case in which we do not impose any stability condition.  In addition, those results in which we do impose a genericity assumption on $\zeta$ apply to the case in which we impose no stability condition and $Q$ is symmetric, since in this case maximally degenerate stability conditions are still generic in our sense. 

Let $\zeta',\zeta''\in\mathbb{H}^{Q_0}_+$ be a pair of King stability conditions, and let $\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\mathbb{N}^{Q_0}$ be a pair of dimension vectors.  We define ${\mathfrak{M}}^{(\zeta',\zeta'')\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}
\cong X^{(\zeta',\zeta'')\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ to be the stack of short exact sequences $$\begin{aligned}
\label{aseseq}
0\rightarrow \rho'\rightarrow\rho\rightarrow\rho''\rightarrow 0\end{aligned}$$ in which $\rho'$ is a $\zeta'$-semistable $\mathbb{C}Q$-module of dimension vector $\operatorname{\mathbf{d}}'$ and $\rho''$ is a $\zeta''$-semistable $\mathbb{C}Q$-module of dimension vector $\operatorname{\mathbf{d}}''$, and $X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{(\zeta',\zeta'')\operatorname{-ss}}\subset X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ is the subspace of flags of $\mathbb{C}Q$-modules for which the induced $\operatorname{\mathbf{d}}'$-dimensional module is $\zeta'$-semistable and the induced $\operatorname{\mathbf{d}}''$-dimensional module is $\zeta''$-semistable.  For $\zeta\in\mathbb{H}^{Q_0}_+$ a stability condition, we define ${\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ to be the stack of short exact sequences as in (\[aseseq\]) such that $\rho$ is $\zeta$-semistable.  There is an isomorphism ${\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\cong X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$, where $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}:=X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\cap X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$. If $\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}$ for some $\mu\in(-\infty,\infty)$, there is equality $${\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}={\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{(\zeta,\zeta)\operatorname{-ss}}.$$

Monoidal products {#prodss}
-----------------

In this section we define and introduce the first properties of the categorification of the quantum torus from refined DT theory. 

\[lfDef\] Let $\mathcal{F}\in{\mathcal{D}^{\geq}}(\operatorname{MMHM}(X))$ for $X$ a scheme.  We say that $\mathcal{F}$ is *locally finite* if for each $Z\in \pi_0(X)$ and for each $n\in\mathbb{Z}$, the element $\operatorname{Gr}_W^n(\operatorname{\mathcal{H}}(\mathcal{F})|_Z)$ belongs to ${\mathcal{D}^{b}}(\operatorname{MMHM}(Z))$, where $\operatorname{\mathcal{H}}(\mathcal{F})$ denotes the total cohomology of $\mathcal{F}$, considered as a complex with zero differential, and $\operatorname{Gr}_W^n(\operatorname{\mathcal{H}}(\mathcal{F})|_Z)=0$ for $n\ll 0$. Denote by ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(X))\subset{\mathcal{D}^{\geq}}(\operatorname{MMHM}(X))$ the full subcategory of locally finite objects. 

The category ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ is going to play the role of the category version of the motivic quantum torus of [@KS1].  The definition of this quantum torus is given in [@KS1 Sec.6.2], for now we remark that it is a power series ring in variables $x^{\operatorname{\mathbf{d}}}$ for $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$, and if $\zeta$ is a generic stability condition then the subring of power series in variables $x^{\operatorname{\mathbf{d}}}$ for $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$ is commutative, while whole quantum torus in general is not.  The categorification of this picture, then, should be a monoidal category, for which the monoidal product cannot be upgraded to a symmetric monoidal product, except on fixed subcategories indexed by the slope $\mu$, where a symmetric monoidal structure should be given. 

The moduli scheme ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$ carries a symmetric monoidal structure given by the direct sum, and the product ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\times {\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\xrightarrow{\oplus}{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$ is a finite map of schemes (see [@DaMe4], and [@Meinhardt14 Lem.2.1]).  By Remark \[convProdDef\] the category ${\mathcal{D}^{\geq}}(\operatorname{MMHM}(\mathcal{M}^{\zeta\operatorname{-ss}}))$ carries a symmetric monoidal product $$\mathcal{F}\boxtimes_{\oplus}\mathcal{G}:=\oplus_*(\pi_1^*\mathcal{F}\otimes\pi_2^*\mathcal{G})$$ where $\otimes$ is defined as in (\[simmon\]).  Since $\oplus$ is finite, and $\boxtimes$ is biexact and preserves the weight filtration, it follows that $\boxtimes_{\oplus}$ is biexact and preserves the weight filtration.  The monoidal unit is $\mathbb{Q}_{{{\mathcal{M}}_0^{\zeta\operatorname{-ss}}}}$, the constant pure Hodge module supported on ${\mathcal{M}}_0^{\zeta\operatorname{-ss}}\cong \operatorname{pt}$, the unit of the monoid ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$. Using symmetry of the product, we define, following[^1] [@MSS11 Thm.1] the functor $$\begin{aligned}
\operatorname{Sym}^n_{\boxtimes_{\oplus}}\colon&{\mathcal{D}^{\geq}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))\rightarrow {\mathcal{D}^{\geq}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))\\
&\mathcal{F}\mapsto (\oplus_*\mathcal{F}^{\boxtimes n})^{\Sigma_n}\end{aligned}$$ where $\Sigma_n$ is the permutation group on $n$ letters.  We then define $$\begin{aligned}
\label{symdef}
\operatorname{Sym}_{\boxtimes_{\oplus}}\colon&{\mathcal{D}^{\geq}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}\setminus {\mathcal{M}}_0^{\zeta\operatorname{-ss}}))\rightarrow {\mathcal{D}^{\geq}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}))\\ \nonumber
&\mathcal{F}\mapsto\bigoplus_{n\geq 0}\operatorname{Sym}^n_{\boxtimes_{\oplus}}\mathcal{F}.\end{aligned}$$ Then the following lemma follows from the fact that $$\pi_0(\oplus)\colon \pi_0({\mathcal{M}})\times\pi_0({\mathcal{M}})\rightarrow\pi_0({\mathcal{M}})$$ has finite fibers. 

\[lfRem\] The functor (\[symdef\]) restricts to a functor $$\operatorname{Sym}_{\boxtimes_{\oplus}}\colon{\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}\setminus {\mathcal{M}}^{\zeta\operatorname{-ss}}_0))\rightarrow {\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu})).$$

\[symmPure\] The functor $\operatorname{Sym}_{\boxtimes_{\oplus}}$ takes pure objects to pure objects. 

Let $\mathcal{F}\in\operatorname{MMHM}(X)$ and $\mathcal{G}\in\operatorname{MMHM}(Y)$ for two algebraic varieties $X$ and $Y$. We define their external tensor product $$\mathcal{F}\boxtimes \mathcal{G}:=(p\colon X\times Y\times\mathbb{A}^1\times\mathbb{A}^1\xrightarrow{\operatorname{id}_{X\times Y}\times +} X\times Y\times \mathbb{A}^1)_*(\operatorname{pr}_{1,3}^*\mathcal{F}\otimes\operatorname{pr}_{2,4}^*\mathcal{G}).$$ as in Remark \[convProdDef\].  Then by [@KS2 Lem.1], there is an isomorphism in ${\mathcal{D}}(\operatorname{MMHM}(X\times Y))$ $$p_!\left(\operatorname{pr}_{1,3}^*\mathcal{F}\otimes\operatorname{pr}_{2,4}^*\mathcal{G}\right)\rightarrow p_*\left(\operatorname{pr}_{1,3}^*\mathcal{F}\otimes\operatorname{pr}_{2,4}^*\mathcal{G}\right)$$ and so since $p_!\colon {\mathcal{D}}(\operatorname{MHM}(X\times \mathbb{A}^1\times Y\times\mathbb{A}^1))\rightarrow {\mathcal{D}}(\operatorname{MHM}(X\times Y\times \mathbb{A}^1))$ is left exact, and decreases weights, while $p_*$ is right exact and increases weights, we deduce that the external tensor product $$\boxtimes\colon {\mathcal{D}}(\operatorname{MMHM}(X))\times{\mathcal{D}}(\operatorname{MMHM}(Y))\rightarrow {\mathcal{D}}(\operatorname{MMHM}(X\times Y))$$ is exact and preserves pure objects.  Let $\mathcal{F}\in\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})$ be pure.  Let $$\oplus^n\colon {\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\times\ldots\times {\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$$ be the n-fold monoid map, then $\oplus^n$ is finite by [@Meinhardt14 Lem.2.1], and so $$\mathcal{G}=\oplus^n_*\left(\mathcal{F}\boxtimes\ldots \boxtimes\mathcal{F}\right)\in\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})$$ is pure.  On the other hand, $\operatorname{Sym}_{\boxtimes_{\oplus}}^n\mathcal{F}\subset \mathcal{G}$ as the isotrivial direct summand under the $\Sigma_n$-action, and is therefore pure.  We deduce that $\operatorname{Sym}_{\boxtimes_{\oplus}}\mathcal{F}$ is a direct sum of pure objects, and is therefore pure. 

We define a new monoidal structure on ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}))$ by setting $$\mathcal{F}\boxtimes_{\oplus}^{\operatorname{tw}} \mathcal{G}:=\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\mathbb{N}^{Q_0}}\oplus_*(\pi_1^*\mathcal{F}_{\operatorname{\mathbf{d}}'}\otimes\pi_2^*\mathcal{G}_{\operatorname{\mathbf{d}}''})\otimes{\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2},$$ where $\mathcal{F}=\bigoplus_{\operatorname{\mathbf{d}}'\in\mathbb{N}^{Q_0}}\mathcal{F}_{\operatorname{\mathbf{d}}'}$ for $\mathcal{F}_{\operatorname{\mathbf{d}}'}\in{\mathcal{D}}(\operatorname{MMHM}({\mathcal{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}))$ and $\mathcal{G}=\bigoplus_{\operatorname{\mathbf{d}}''\in\mathbb{N}^{Q_0}}\mathcal{G}_{\operatorname{\mathbf{d}}''}$ for $\mathcal{G}_{\operatorname{\mathbf{d}}''}\in{\mathcal{D}}(\operatorname{MMHM}({\mathcal{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}))$. If $\zeta$ is $\mu$-generic, the restriction of $\boxtimes_{\oplus}^{\operatorname{tw}}$ to ${\mathcal{D}^{\geq}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}))$ is naturally isomorphic to the untwisted symmetric monoidal product $\boxtimes_{\oplus}$, but in general there is no natural isomorphism of bifunctors making $\boxtimes_{\oplus}^{\operatorname{tw}}$ into a symmetric monoidal product. 

Similarly, we define the monoidal product on ${\mathcal{D}^{\geq}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ by setting $$\mathcal{F}_{\operatorname{\mathbf{d}}'}\boxtimes_+^{\operatorname{tw}}\mathcal{G}_{\operatorname{\mathbf{d}}''}:=+_*(\mathcal{F}_{\operatorname{\mathbf{d}}'}\boxtimes\mathcal{G}_{\operatorname{\mathbf{d}}''})\otimes{\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2},$$ where $\mathcal{F}_{\operatorname{\mathbf{d}}'}$ is a monodromic mixed Hodge module with support $\operatorname{\mathbf{d}}'\in\mathbb{N}^{Q_0}$ and $\mathcal{G}_{\operatorname{\mathbf{d}}''}$ is a monodromic mixed Hodge module with support $\operatorname{\mathbf{d}}''$. This prescription extends to a unique monoidal product $\boxtimes_+^{\operatorname{tw}}$ that commutes with arbitrary direct sums, and turns the map $$\dim_*\colon {\mathcal{D}^{\geq}}(\operatorname{MMHM}(\mathcal{M}^{\zeta\operatorname{-ss}}))\rightarrow {\mathcal{D}^{\geq}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$$ into a monoidal functor, where the domain is given the twisted monoidal structure $\boxtimes^{\operatorname{tw}}_{\oplus}$.

\[twDual\] By skew symmetry of $\langle \bullet,\bullet\rangle_Q$ and finiteness of the maps of schemes $\oplus$ and $+$, there are natural isomorphisms of bifunctors $$\begin{aligned}
&{\mathbb{D}}^{\operatorname{mon}}(\mathcal{F}\boxtimes_{\oplus}^{\operatorname{tw}}\mathcal{G})\cong {\mathbb{D}}^{\operatorname{mon}}\mathcal{G}\boxtimes_{\oplus}^{\operatorname{tw}}{\mathbb{D}}^{\operatorname{mon}}\mathcal{F}\\
&{\mathbb{D}}^{\operatorname{mon}}(\mathcal{F}\boxtimes_{+}^{\operatorname{tw}}\mathcal{G})\cong {\mathbb{D}}^{\operatorname{mon}}\mathcal{G}\boxtimes_{+}^{\operatorname{tw}}{\mathbb{D}}^{\operatorname{mon}}\mathcal{F}.\end{aligned}$$ Note the swap of arguments. 

Framed moduli spaces {#framedSec}
--------------------

Framed moduli spaces will play a central role in what follows.  Their cohomology provides an approximation to the cohomology of ${\mathfrak{M}}^{\zeta\operatorname{-ss}}$, in a way which we will make precise in Section \[cohaDT\]. 

Let ${\mathbf{f}}\in\mathbb{N}^{Q_0}$ be a dimension vector (called the framing vector).  We form a new quiver $Q_{{\mathbf{f}}}$ by setting $Q_{{\mathbf{f}}}=(Q_0\sqcup\{\infty\}, Q_1\sqcup \{\beta_{i,l_i}\colon \infty \to i \mid i\in Q_0, \hbox{ }1\le l_i\le {\mathbf{f}}_i \})$. Given a stability condition $\zeta$ for $Q$, and a slope $\mu\in(-\infty,\infty)$, and a $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$ we extend $\operatorname{\mathbf{d}}$ to a dimension vector for $Q_{{\mathbf{f}}}$ by setting $\operatorname{\mathbf{d}}_{\infty}=1$, and we extend $\zeta$ to a stability condition $\zeta^{(\mu)}_{{\mathbf{f}}}$ for $Q_{{\mathbf{f}}}$ by picking $\zeta^{(\mu)}_{{\mathbf{f}},\infty}\in\mathbb{H}_+$ so that $$-\Re e(\zeta^{(\mu)}_{{\mathbf{f}},\infty})/\Im m(\zeta^{(\mu)}_{{\mathbf{f}},\infty})=\mu+\epsilon$$ for sufficiently small $\epsilon>0$, and picking $|\zeta_{{\mathbf{f}},\infty}^{(\mu)}|\gg 0$.

A $\mathbb{C} Q_{{\mathbf{f}}}$-module $\rho$ with $\dim(\rho)_{\infty}=1$ is $\zeta^{(\mu)}_{{\mathbf{f}}}$-semistable if and only if the underlying $\mathbb{C} Q$-module is $\zeta$-semistable, and for every $\mathbb{C}Q_{{\mathbf{f}}}$-submodule $\rho'\subset \rho$ such that $\dim(\rho')_{\infty}=1$, the underlying $\mathbb{C} Q$-module of $\rho'$ has slope strictly less than $\mu$. A $\zeta^{(\mu)}_{{\mathbf{f}}}$-semistable $\mathbb{C}Q_{{\mathbf{f}}}$-module is automatically $\zeta^{(\mu)}_{{\mathbf{f}}}$-stable, and we write ${\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ for the coarse moduli space of $\zeta^{(\mu)}_{{\mathbf{f}}}$-semistable $\mathbb{C} Q_{{\mathbf{f}}}$-modules of dimension $(1,\operatorname{\mathbf{d}})\in\mathbb{N}\times\mathbb{N}^{Q_0}$. The moduli space ${\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ is smooth. 

In fact $G_{\operatorname{\mathbf{d}}}$ acts freely on the variety $$\label{Ydef}
Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}:=X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})},$$ and ${\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ is the quotient.  From this description we deduce that ${\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ is a smooth ${\mathbb{G}}_m$-torsor over $Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{(1,\operatorname{\mathbf{d}})}$.

We denote by $$\pi_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\colon{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\rightarrow {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$$ the map given by forgetting the framing.  It is a proper map, since the other two maps in the diagram $$\xymatrix{
{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\ar@/^1pc/[rr]\ar[r]_-{\pi^\zeta_{{\mathbf{f}},\operatorname{\mathbf{d}}}}&{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\ar[r]_-{q^\zeta_{\operatorname{\mathbf{d}}}}&{\mathcal{M}}_{\operatorname{\mathbf{d}}}
}$$ are, as they are GIT quotient maps.  For $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$ we may alternatively extend $\operatorname{\mathbf{d}}$ to a dimension vector for $Q_{{\mathbf{f}}}$ by setting $\operatorname{\mathbf{d}}_{\infty}=0$. There is a natural isomorphism $X(Q_{{\mathbf{f}}})_{(0,\operatorname{\mathbf{d}})}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}\cong X(Q)_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$. Later we will use a slightly different stability condition on framed modules.  We define $\zeta_{{\mathbf{f}}}^{[\mu]}$ by picking $-\Re e(\zeta^{[\mu]}_{{\mathbf{f}},\infty})/\Im m(\zeta^{[\mu]}_{{\mathbf{f}},\infty})=\mu$. For definiteness, we pick $\lvert \zeta_{{\mathbf{f}}}^{[\mu]}\rvert=1$. A $\mathbb{C}Q_{{\mathbf{f}}}$-module is $\zeta_{{\mathbf{f}}}^{[\mu]}$-semistable if and only if the underlying $\mathbb{C}Q$-module is $\zeta$-semistable.  As such, there is a natural inclusion of stacks (in which the domain is a scheme) $$\label{jmap}
{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}:=Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}}\xrightarrow{j_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}} X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{[\mu]}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}/G_{\operatorname{\mathbf{d}}}.$$ Note that if $Q$ is acyclic then ${\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ is proper, but the map (\[jmap\]) may still be profitably thought of as a partial compactification of ${\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ in the category of stacks. 

Jacobi algebras and potentials
------------------------------

Let $W\in\mathbb{C} Q/[\mathbb{C}Q,\mathbb{C}Q]$ be a finite linear combination of equivalence classes of cyclic words in $\mathbb{C} Q$ – such a $W$ is called a *potential*.  A potential $W$ induces a function ${\mathfrak{Tr}(W)}$ on ${\mathfrak{M}}$, defined as follows.  Firstly, assume that $W$ lifts to a single cyclic word $c=a_r\ldots a_0$ in $\mathbb{C} Q$. Then a $\mathbb{C}Q$-module $F$ determines an endomorphism $$F(a_r)\circ\ldots\circ F(a_0)\colon F(s(a_0))\rightarrow F(s(a_0)),$$ and $\operatorname{Tr}(F(a_r)\circ\ldots\circ F(a_0))$ determines a function on $X(Q)_{\operatorname{\mathbf{d}}}$, for each $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$. By cyclic invariance of the trace, this function is $G_{\operatorname{\mathbf{d}}}$-invariant, and does not depend on the lift $c$ of $W$. Extending by linearity, we define for general $W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]$ the induced function $${\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}\colon {\mathfrak{M}}_{\operatorname{\mathbf{d}}}\rightarrow \mathbb{C}.$$ We denote by ${\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}^{\zeta}$ the restriction of this function to ${\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$, and by $${\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}\colon {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\rightarrow\mathbb{C}$$ the unique function through which ${\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}$ factors.  Similarly, we define $${\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}:={\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}\circ\pi_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\colon {\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\rightarrow \mathbb{C},$$ and we define ${\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}$ to be the composition $${\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\hookrightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\xrightarrow{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''}^{\zeta}}\mathbb{C}.$$

Associated to the data $(Q,W)$ is the Jacobi algebra $$\operatorname{Jac}(Q,W):=\mathbb{C} Q/\langle \partial W/\partial a|a\in Q_1\rangle.$$ Here the *noncommutative derivative* $\partial W/\partial a$ is defined as follows.  First assume that $W$ lifts to a single cyclic word $c\in\mathbb{C}Q$. Then $$\partial W/\partial a:=\sum_{c=c'ac''}c''c'.$$ We then extend the definition to general $W$ by linearity.  We define ${\mathfrak{M}}_{W}$, the stack of finite-dimensional $\operatorname{Jac}(Q,W)$-modules, in the same way as the stack of finite-dimensional $\mathbb{C} Q$-modules.  In particular there is a natural closed embedding of stacks ${\mathfrak{M}}_W\subset {\mathfrak{M}}$, and it is easy to show that ${\mathfrak{M}}_W=\operatorname{crit}({\mathfrak{Tr}(W)})$ as substacks of ${\mathfrak{M}}$. In order to keep to Assumption \[phicheat\] we will assume that $W\in \langle \partial W/\partial a|a\in Q_1\rangle$, for then $\operatorname{crit}({\mathfrak{Tr}(W)})\subset{\mathfrak{Tr}(W)}^{-1}(0)$. One very common set of circumstances in which this requirement is met is when there is a grading of the arrows $Q_1$ with integers such that $W$ is homogeneous of nonzero weight.  As mentioned after Assumption \[phicheat\], we can drop this requirement at the expense of slightly more complicated definitions. 

\[TSeq\] Let $W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]$ be a potential.  Then $${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\colon\operatorname{MHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu})\rightarrow\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu})$$ is a symmetric monoidal functor. 

Let $\mathcal{F}\in\operatorname{MHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu})$. The statement follows from the claim that the following diagram is commutative, $$\xymatrix{
{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\left(\mathcal{F}\boxtimes_{\oplus}\mathcal{F}\right)\ar[r]\ar[d]&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\left(\mathcal{F}\boxtimes_{\oplus}\mathcal{F}\right)\ar[d]\\
{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\mathcal{F}\boxtimes_{\oplus}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\mathcal{F}\ar[r]&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\mathcal{F}\boxtimes_{\oplus}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\mathcal{F},
}$$ where the horizontal maps are induced by the symmetric monoidal structure, and the vertical maps are the Thom-Sebastiani isomorphism.  By faithfulness of the functor ${\operatorname{forg}^{\operatorname{mon}}_{{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}}}$ defined in (\[formDef\]), it is enough to prove commutativity of the diagram at the level of perverse sheaves.  Let $q\colon {\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\times{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\rightarrow \operatorname{Sym}^2({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})$ be the natural quotient map, and let $r\colon \operatorname{Sym}^2({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})\rightarrow{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}$ be the map induced by $\oplus$. Then $r$ is finite, and by natural commutativity of vanishing cycle functors with proper maps, it is enough to show that the diagram $$\xymatrix{
\phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\left(q_*\left(\mathcal{F}\boxtimes\mathcal{F}\right)\right)\ar[r]\ar[d]&\phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\left(q_*\left(\mathcal{F}\boxtimes\mathcal{F}\right)\right)\ar[d]\\
q_*\left(\phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\mathcal{F}\boxtimes\phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\mathcal{F}\right)\ar[r]&q_*\left(\phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\mathcal{F}\boxtimes_{\oplus}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\mathcal{F}\right),
}$$ commutes, or equivalently, that the Thom–Sebastiani isomorphism $$\phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\mathcal{F}\boxtimes \phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}\mathcal{F}\rightarrow \phi_{{\mathcal{T}r(W)}_{\mu}^{\zeta}\boxplus {\mathcal{T}r(W)}_{\mu}^{\zeta}}\left(\mathcal{F}\boxtimes \mathcal{F}\right)$$ commutes with the $\Sigma_2$-equivariant structures on both sheaves.  Let $\iota\colon{\mathcal{M}}_{\mu}^{\zeta}\rightarrow X$ be an embedding inside a smooth scheme, let $f$ be a function on $X$ extending ${\mathcal{T}r(W)}_{\mu}^{\zeta}$ and consider $\mathcal{F}$ as a perverse sheaf on $X$ via the direct image.  Then we define the functor on $\operatorname{MHM}(X)$ $$\Gamma_{f^{-1}(\mathbb{R}_{\leq 0})}\mathcal{F}(U):=\ker\left(\mathcal{F}(U)\rightarrow \mathcal{F}(U\setminus f^{-1}(\mathbb{R}_{\leq 0}))\right)$$ and we may alternatively define $\phi_f\mathcal{F}:=(R\Gamma_{(f^{-1}(\mathbb{R}_{\leq 0})}\mathcal{F})|_{f^{-1}(0)}$. By [@Br12 Prop. A.2] [@Ma01] the Thom–Sebastiani isomorphism is given by the natural map $$\alpha\colon R\Gamma_{f^{-1}(\mathbb{R}_{\leq 0})\times f^{-1}(\mathbb{R}_{\leq 0})}\left(\mathcal{F}\boxtimes\mathcal{F}\right)\rightarrow R\Gamma_{(f\boxplus f)^{-1}(\mathbb{R}_{\leq 0})}\left(\mathcal{F}\boxtimes\mathcal{F}\right)$$ induced by the inclusion $$f^{-1}(\mathbb{R}_{\leq 0})\times f^{-1}(\mathbb{R}_{\leq 0})\subset (f\boxplus f)^{-1}(\mathbb{R}_{\leq 0}),$$ and the $\Sigma_2$-equivariant structure is given by the $\Sigma_2$-equivariant structure on $\mathcal{F}\boxtimes\mathcal{F}$ on both sides, and so $\alpha$ lifts to a morphism of $\Sigma_2$-equivariant perverse sheaves. 

Cohomological Donaldson–Thomas invariants
=========================================

Proof of Theorem \[ThmA\] {#cohaDT}
-------------------------

We start this section by showing that there is a module-theoretic partial compactification of Totaro’s construction, which is what we used in Section \[TotCon\] to define the direct image of the monodromic mixed Hodge module of vanishing cycles from the stack of representations of $Q$.

Recall the definition of $\zeta_{{\mathbf{f}}}^{(\mu)}$ from Section \[framedSec\].  Recall from (\[Ydef\]) the notation $$\begin{aligned}
X_{{\mathbf{f}},\operatorname{\mathbf{d}}}:=&X(Q_{{\mathbf{f}}})_{(1,\operatorname{\mathbf{d}})}\\
Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}:=&X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}.\end{aligned}$$ Assume that $\zeta_{{\mathbf{f}}}^{(\mu)}$ is a King stability condition for $Q_{{\mathbf{f}}}$, as we always can, by Remark \[BtoK\].  Then $\zeta_{{\mathbf{f}}}^{(\mu)}$ defines a linearization of the natural $G_{(1,\operatorname{\mathbf{d}})}$-action on $X_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ and we have $$\begin{aligned}
{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}:=&X_{{\mathbf{f}},\operatorname{\mathbf{d}}}/\!\!/_{\chi}G_{(1,\operatorname{\mathbf{d}})}
\\\cong &Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}}\end{aligned}$$ where $G_{(1,\operatorname{\mathbf{d}})}=\operatorname{GL}_1(\mathbb{C})\times G_{\operatorname{\mathbf{d}}}$. As a $G_{\operatorname{\mathbf{d}}}$-equivariant variety, $X_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ admits a product decomposition $X_{{\mathbf{f}},\operatorname{\mathbf{d}}}=X_{\operatorname{\mathbf{d}}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}}$, where $V_{{\mathbf{f}},\operatorname{\mathbf{d}}}:=\bigoplus_{i\in Q_0}\operatorname{Hom}(\mathbb{C}^{{\mathbf{f}}_i},\mathbb{C}^{\operatorname{\mathbf{d}}_i})$, and the extra $\operatorname{GL}_1(\mathbb{C})$ factor acts by rescaling $V_{{\mathbf{f}},\operatorname{\mathbf{d}}}$. We define $U_{{\mathbf{f}},\operatorname{\mathbf{d}}}\subset V_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ to be the subspace for which each $\alpha_i\in\operatorname{Hom}(\mathbb{C}^{{\mathbf{f}}_i},\mathbb{C}^{\operatorname{\mathbf{d}}_i})$ is surjective.  The group $G_{\operatorname{\mathbf{d}}}$ acts freely on $U_{{\mathbf{f}},\operatorname{\mathbf{d}}}$, and the quotient is a product of Grassmannians, and exists as a fibre bundle quotient in the category of schemes.  We have the following commutative diagram $$\xymatrix{
\\
(X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times U_{{\mathbf{f}},\operatorname{\mathbf{d}}})/G_{\operatorname{\mathbf{d}}}\ar@/^2pc/[rr]^-{i}\ar[drr]_{\kappa_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\ar@{^(->}[r]^-h& Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}}\ar@{^(->}[r]\ar[rd]^{\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}& (X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}})/G_{\operatorname{\mathbf{d}}}\ar[d]
\\&& {\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}. 
}$$ All of the objects in the above diagram are algebraic varieties, with the exception of $(X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}})/G_{\operatorname{\mathbf{d}}}$, which is an Artin stack.  The diagram obtained by deleting this stack and all arrows incident to it is a relative compactification of the map $\kappa_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$.

In what follows we use the notation ${\mathbf{f}}\gg 0$ to mean that ${\mathbf{f}}_i\gg 0$ for every $i\in Q_0$.

\[appLem\] For fixed $n$ and ${\mathbf{f}}\gg 0$ the natural map $$\operatorname{\mathcal{H}}^n\left(\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}}}\right)\rightarrow\operatorname{\mathcal{H}}^n\left(\kappa^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{(X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times U_{{\mathbf{f}},\operatorname{\mathbf{d}}})/G_{\operatorname{\mathbf{d}}}}\right)$$ is an isomorphism. 

As in the proof of Proposition \[stabProp\] it is enough to show that for fixed $n\in\mathbb{Z}$, $\operatorname{\mathcal{H}}_{\operatorname{con}}^n\left(\phi_{\overline{{\mathcal{T}r(W)}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\mathbb{Q}_{Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\rightarrow\overline{h}_*\overline{h}^*\phi_{\overline{{\mathcal{T}r(W)}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\mathbb{Q}_{Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)$ is an isomorphism for ${\mathbf{f}}\gg 0$, where $$\overline{h}\colon X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times U_{{\mathbf{f}},\operatorname{\mathbf{d}}} \rightarrow Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$$ is the inclusion and $$\overline{{\mathcal{T}r(W)}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\colon Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\rightarrow \mathbb{C}$$ is the composition of ${\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ with the quotient map.  On the other hand, the complex of sheaves $\phi_{\overline{{\mathcal{T}r(W)}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\mathbb{Q}_{Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}$ is given by restriction of the pullback $\rho^*\phi_{\overline{{\mathcal{T}r(W)}}_{\operatorname{\mathbf{d}}}}\mathbb{Q}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}$ of $\phi_{\overline{{\mathcal{T}r(W)}}_{\operatorname{\mathbf{d}}}}\mathbb{Q}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}$ along the projection $\rho\colon X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}}\rightarrow X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$. It follows that the cohomology (with respect to the constructible t structure) of the complex of sheaves $\phi_{\overline{{\mathcal{T}r(W)}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\mathbb{Q}_{Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}$ vanishes outside of a range that is independent of ${\mathbf{f}}$, and the result follows from the fact that as ${\mathbf{f}}\gg 0$, the codimensions of $\left(X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}}\setminus Y^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\right)$ and $\left(X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}}\setminus X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\times U_{{\mathbf{f}},\operatorname{\mathbf{d}}}\right)$ inside $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\times V_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ go to infinity. 

We deduce that for fixed $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$ and $n\in\mathbb{Z}$, and for ${\mathbf{f}}\gg 0$ there are isomorphisms $$\label{PhiSurj}
\Phi_{{\mathbf{f}},\operatorname{\mathbf{d}},W}\colon \operatorname{\mathcal{H}}^n(\pi _{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}})\rightarrow \operatorname{\mathcal{H}}^n(\kappa^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{(X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times U_{{\mathbf{f}},\operatorname{\mathbf{d}}})/G_{\operatorname{\mathbf{d}}}})$$ and $$\label{PsiSurj}
\Psi_{{\mathbf{f}},\operatorname{\mathbf{d}},W}\colon  \operatorname{\mathcal{H}}^n(\kappa^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{(X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times U_{{\mathbf{f}},\operatorname{\mathbf{d}}})/G_{\operatorname{\mathbf{d}}}}\otimes\mathbb{L}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}})\rightarrow \operatorname{\mathcal{H}}^n(\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes\mathbb{L}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}})$$ where the argument that $\Psi_{{\mathbf{f}},\operatorname{\mathbf{d}},W}$ is an isomorphism is as in Lemma \[appLem\].  On the other hand, for ${\mathbf{f}}\gg 0$, the right hand side of (\[PhiSurj\]) is by definition $\operatorname{\mathcal{H}}^n(p_{\operatorname{\mathbf{d}},*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}})$, while the left hand side of (\[PsiSurj\]) is $\operatorname{\mathcal{H}}^n(p_{\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}})$.

Put in words, we can say that the direct image of the vanishing cycle monodromic mixed Hodge module along the non-representable map $p^{\zeta}_{\operatorname{\mathbf{d}}}\colon{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\rightarrow{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$ is approximated by the direct image along the representable and *proper* maps $\pi _{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\colon {\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\rightarrow{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$.

For ${\mathbf{f}}\gg 0$ we obtain isomorphisms $$\label{PhiBar}
\overline{\Phi}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\colon \operatorname{H}^n({\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}})\cong\operatorname{H}^n({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}})$$ and $$\label{PsiBar}
\overline{\Psi}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\colon \operatorname{H}^n_c({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}})\cong \operatorname{H}^n_c({\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}})$$ in the same way. 

Since each map $\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}$ is proper, and ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}$ is exact, we deduce the following proposition. 

\[cvs\] For every $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$ there are isomorphisms $$\label{nudef}
\nu_{\operatorname{\mathbf{d}}}\colon \operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}})\cong {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}).$$

The isomorphisms $\nu_{\operatorname{\mathbf{d}}}$ are obtained by considering the left hand side of (\[PhiSurj\]) and using the natural isomorphisms $${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\cong \pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}.$$ The isomorphisms $\nu_{\operatorname{\mathbf{d}}}$ are well-defined by commutativity of the square $$\xymatrix{
\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}}\ar[r]&\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}\iota_{{\mathbf{f}},{\mathbf{f}}',*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\\
{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}}\ar[u]\ar[r]& {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}\iota_{{\mathbf{f}},{\mathbf{f}}',*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\ar[u]
}$$ where ${\mathbf{f}}'>{\mathbf{f}}$ and $\iota_{{\mathbf{f}},{\mathbf{f}}'}\colon{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\rightarrow {\mathcal{M}}^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}$ is the natural inclusion obtained by extending a framing by zero.  The square commutes as it is obtained by applying the natural transformation $${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}\rightarrow \pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}}}$$ to the restriction map $$\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}}\rightarrow \iota_{{\mathbf{f}},{\mathbf{f}}',*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}.$$

Similarly, the natural isomorphisms $ {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\cong \pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}$, along with exactness of ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}$, induce natural isomorphisms $$\nu_{c,\operatorname{\mathbf{d}}}\colon \operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}},!}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}})\cong{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\operatorname{\mathcal{H}}(p_{\operatorname{\mathbf{d}},!}^{\zeta}\mathbb{Q}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}).$$ Now assume that $\zeta$ is a $\mu$-generic Bridgeland stability condition.  As in [@DaMe4] and the introduction, for $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}\setminus \{0\}$ we define the following element of $\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}})$ $${\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}=\begin{cases} {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^\zeta_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta \operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})&\textrm{if }{\mathcal{M}}^{\zeta \operatorname{-st}}_{\operatorname{\mathbf{d}}}\neq\emptyset ,\\0&\textrm{otherwise.}\end{cases}$$ We define $${\mathcal{DT}}^{\zeta}_{W,\mu}:=\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}\setminus \{0\}}{\mathcal{DT}}^{\zeta}_{W,\operatorname{\mathbf{d}}}\in\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}).$$

Note that from our shifting convention on ${\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})$, along with exactness of ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}$ it follows that ${\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}$ is indeed a genuine (monodromic) mixed Hodge module, instead of merely being an element of ${\mathcal{D}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}))$. One can find examples for which it is not pure.  For example consider the quiver $Q$ with three loops and potential $W=x^p+y^q+z^r+axyz$, with $a\neq 0$ and $p^{-1}+q^{-1}+r^{-1}<1$. Then ${\mathcal{M}}_1=\mathbb{A}^3$, and on ${\mathcal{M}}_1$ there is an identity ${\mathrm{Tr}(W)}=W$, and the potential has an isolated singularity at the origin.  The cohomology $\operatorname{\mathcal{H}}({\phi^{\operatorname{mon}}_{W}}\mathbb{Q}_{\mathbb{A}^3})$ is not pure (see [@DMSS15 Rem.3.5] [@Kul98 Ex.7.3.5] [@Sch85 Ex.9.1]), and so $${\mathcal{DT}}_{W,1}:=\operatorname{\mathcal{H}}({\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{1}}}\mathbb{Q}_{{\mathcal{M}}_1})\otimes\mathbb{L}^{-3/2}$$ is not pure either. 

\[sdrem\] Using Proposition \[basicfacts\] we have isomorphisms $$\begin{aligned}
{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\cong& {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\\
\cong&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\end{aligned}$$ and so $$\label{sdual}
{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}{\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}\cong{\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}.$$

\[abbrev\] We make the abbreviation of symbols $${\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta}:={\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}(\mathbb{Q})$$ and $${\mathfrak{IC}}_{W,\mu}^{\zeta}:={\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\mu}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\mu}^{\zeta\operatorname{-ss}}}(\mathbb{Q}).$$

\[weakPBW\] Assume that $\zeta$ is a $\mu$-generic stability condition on the quiver $Q$. There is an isomorphism in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}))$ $$\label{weakeq}
\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta})\cong\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)$$ and an isomorphism in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\Lambda_{\mu}^{\zeta}))$ $$\label{vweakeq}
\operatorname{\mathcal{H}}(\dim_*p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta})\cong\operatorname{Sym}_{\boxtimes_{+}}\left(\operatorname{\mathcal{H}}(\dim_*{\mathcal{DT}}_{W,\mu}^{\zeta})\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right).$$

By definition, the left hand side of (\[weakeq\]) is isomorphic to its total cohomology.  On the other hand, $\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$ is also isomorphic to its total cohomology, and ${\mathcal{DT}}_{W,\mu}^{\zeta}$ is an object of $\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})\subset {\mathcal{D}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))$, and is trivially isomorphic to its total cohomology.  It follows from the exactness of $\boxtimes_{\oplus}$ that the right hand side of (\[weakeq\]) is also isomorphic to its total cohomology, and so it is sufficient to construct the isomorphism (\[weakeq\]) at each cohomological degree. 

We first show the result, under the assumption that $W=0$. That is, we show that $\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}}(\mathbb{Q}))\cong\operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{W=0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})$. By Lemma \[appLem\], for fixed $n\in\mathbb{Z}$ and ${\mathbf{f}}\gg 0$ the map $$\operatorname{\mathcal{H}}^n\left(\pi _{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}(\mathbb{Q})\otimes {\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2}\right)\rightarrow\operatorname{\mathcal{H}}^n\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{0,\operatorname{\mathbf{d}}}^{\zeta}\right)$$ is an isomorphism.  It follows that $\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{0,\mu}^{\zeta})$ is pure, since $\operatorname{\mathcal{H}}^n(\pi _{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}(\mathbb{Q}))$ is a pure mixed Hodge module of weight $n$, as purity is preserved by direct image along proper maps.  It follows that $\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{0,\mu})$ is locally finite in the sense of Definition \[lfDef\], as is $\operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})$, by Lemma \[lfRem\], since $${\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\in{\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\setminus{\mathcal{M}}^{\zeta\operatorname{-ss}}_0)).$$ The element $\operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})$ is also a pure monodromic mixed Hodge module, by Proposition \[symmPure\], and so both $\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}}(\mathbb{Q}))$ and $\operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})$ are direct sums of simple pure mixed Hodge modules, and are isomorphic if and only if they have the same class in $\operatorname{K}_0({\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu})))$. From the fact that the map $\Phi^n_{0,{\mathbf{f}},\operatorname{\mathbf{d}}}$ of equation (\[PhiBar\]) is an isomorphism for fixed $n$ and ${\mathbf{f}}\gg 0$ we deduce that $$\begin{aligned}
\left[\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}}(\mathbb{Q})\right)\right]_{\operatorname{K}_0}=&\lim_{{\mathbf{f}}\mapsto \infty}[\pi _{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\otimes{\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2}]_{\operatorname{K}_0}\\=&\lim_{{\mathbf{f}}\mapsto\infty}\left[\operatorname{Sym}_{\boxtimes_{\oplus}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\mathcal{DT}}_{0,\operatorname{\mathbf{d}}}^{\zeta}\otimes\operatorname{H}(\mathbb{P}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}-1})_{\operatorname{vir}}\otimes{\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2}\right)\right]_{\operatorname{K}_0}\\=&\left[\operatorname{Sym}_{\boxtimes_{\oplus}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\mathcal{DT}}^{\zeta}_{0,\operatorname{\mathbf{d}}}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right]_{\operatorname{K}_0}\end{aligned}$$ as required – for the second equality we have used the main result of [@Meinhardt14].  For the case $W\neq 0$, by Proposition \[TSeq\] we deduce the existence of isomorphisms $$\begin{aligned}
\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}}(\mathbb{Q})\right)&\cong^{\nu_{\operatorname{\mathbf{d}}}}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\mu}}(\mathbb{Q})\right)\\
&\cong {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}\operatorname{Sym}_{\boxtimes_{\oplus}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\mathcal{DT}}^{\zeta}_{0,\operatorname{\mathbf{d}}}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\\
&\cong \operatorname{Sym}_{\boxtimes_{\oplus}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{DT}}^{\zeta}_{0,\operatorname{\mathbf{d}}}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\\
&= \operatorname{Sym}_{\boxtimes_{\oplus}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\mathcal{DT}}^{\zeta}_{W,\operatorname{\mathbf{d}}}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right).\end{aligned}$$ Finally, for the absolute case of (\[vweakeq\]), we use the following chain of isomorphisms, for ${\mathbf{f}}\gg 0$, where the second is the decomposition theorem for the direct image of the vanishing cycles along a proper map from Proposition \[basicfacts\](\[vanDec\]): $$\begin{aligned}
\operatorname{\mathcal{H}}^n\left(\dim_*p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right) &\cong \operatorname{\mathcal{H}}^n\left(\dim_*\pi _{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\otimes{\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2}\right)\\
&\cong \operatorname{\mathcal{H}}^n\left(\dim_*\operatorname{\mathcal{H}}(\pi _{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\otimes{\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2})\right)\\
&\cong \operatorname{\mathcal{H}}^n\left(\dim_*\operatorname{Sym}_{\boxtimes_{\oplus}}\left(\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\mathcal{DT}}^{\zeta}_{W,\operatorname{\mathbf{d}}}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\\
&\cong \operatorname{Sym}_{\boxtimes_{+}}\left(\operatorname{\mathcal{H}}(\dim_*{\mathcal{DT}}_{W,\mu}^{\zeta})\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right).\end{aligned}$$

\[weakPBWrem\] We can think of Theorem \[weakPBW\] as a ‘weak’ Poincaré–Birkhoff-Witt theorem for $\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{W,\mu}\right)$. It is a PBW theorem in the sense that it asserts that as an object in a symmetric monoidal category, $\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{W,\mu}\right)$ is isomorphic to a free symmetric algebra.  It is ‘weak’ in the sense that the isomorphism itself is not specified in terms of an algebra structure on $\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{W,\mu}\right)$. In Section \[CoHAsec\] we will introduce an algebra structure on $\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{W,\mu}\right)$ relativizing the Kontsevich–Soibelman construction [@KS2 Sec.7] over the base ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$, and prove the corresponding ‘strong’ PBW theorem (Theorem \[strongPBW\]) in Section \[PBWsec\]. 

Taking the Verdier dual of both sides of (\[weakeq\]) we deduce the following corollary. 

Assume that $\zeta$ is $\mu$-generic.  Then there is an isomorphism $$\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,!}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\cong\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})^{\vee}_{\operatorname{vir}}\right)$$ where $\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})^{\vee}_{\operatorname{vir}}:=\operatorname{\mathcal{H}om}_{\operatorname{MMHM}(\operatorname{pt})}(\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}},\mathbb{Q})$ is the dual monodromic mixed Hodge structure. 

\[inccor\] There is a canonical inclusion $$\label{caninc}
{\mathcal{DT}}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\rightarrow \operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)$$

First we prove the corollary in the case $W=0$. In this case, the left hand side of (\[caninc\]) is precisely the summand of the right hand side with strict supports equal to ${\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$ for $\operatorname{\mathbf{d}}\in\Lambda^{\zeta}_{\mu}$ by Theorem \[weakPBW\], from which the result follows from the decomposition theorem.  For the general case, we apply the functor ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}$ to the inclusion from the case $W=0$.

Cohomological wall crossing formula {#cwcfSec}
-----------------------------------

In this section we prove a weak (in the sense of Remark \[weakPBWrem\]) version of Theorem \[strongPBW\].  We will return to the cohomological wall crossing formula in Section \[CWCF\] once we have introduced the cohomological Hall algebras $\operatorname{\mathcal{H}}(\mathcal{A}_{W,\mu}^{\zeta})$ and $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$, but for the time being we can still prove that there is *some* isomorphism (\[gthmd\]), defined without reference to the cohomological Hall algebra.  The mere existence of this isomorphism is enough for many applications. 

Fix a dimension vector $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$. Denote by $\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$ the set of Harder-Narasimhan types for $\operatorname{\mathbf{d}}$, that is, sequences $\operatorname{\mathbf{d}}^1,\ldots,\operatorname{\mathbf{d}}^s\in\mathbb{N}^{Q_0}$ such that the slopes ${\Xi}^{\zeta}(\operatorname{\mathbf{d}}^1),\ldots,{\Xi}^{\zeta}(\operatorname{\mathbf{d}}^s)$ are strictly decreasing and $\sum_{i=1}^s \operatorname{\mathbf{d}}^i=\operatorname{\mathbf{d}}$. Recall that by [@Reineke_HN Prop.3.4], the moduli stack ${\mathfrak{M}}_{\operatorname{\mathbf{d}}}$ has a stratification by locally closed substacks $$\label{HNstrat}
{\mathfrak{M}}_{\operatorname{\mathbf{d}}}=\coprod_{\overline{\operatorname{\mathbf{d}}}\in\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}}{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}$$ where ${\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}$ is the stack of representations which have Harder–Narasimhan type $\overline{\operatorname{\mathbf{d}}}$ with respect to the stability condition $\zeta$. This is a stratification in the following weak sense.  There is a partial ordering $\leq'$ on $\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$ such that $$\overline{{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}}\subset \bigcup_{\substack{\overline{\operatorname{\mathbf{e}}}\in\operatorname{HN}_{\operatorname{\mathbf{d}}}^{\geq}\\ \overline{\operatorname{\mathbf{e}}}\leq' \overline{\operatorname{\mathbf{d}}}}}{\mathfrak{M}}_{\overline{\operatorname{\mathbf{e}}}}^{\zeta}.$$ Each of the stacks ${\mathfrak{M}}^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}$ can be written as a global quotient stack $${\mathfrak{M}}^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}\cong X^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}/G_{\overline{\operatorname{\mathbf{d}}}}$$ where $X^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}\subset X_{\operatorname{\mathbf{d}}}$ is the subspace of representations preserving the flag defined by $\overline{\operatorname{\mathbf{d}}}$, such that each of the associated $\operatorname{\mathbf{d}}^r$-dimensional subquotients is $\zeta$-semistable, and $G_{\overline{\operatorname{\mathbf{d}}}}\subset G_{\operatorname{\mathbf{d}}}$ is the subgroup preserving the same flag.  Each of these stacks comes with a map $p_{\overline{\operatorname{\mathbf{d}}}}\colon {\mathfrak{M}}^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}\rightarrow{\mathcal{M}}_{\operatorname{\mathbf{d}}}$ sending a representation to its semisimplification, and a representable proper map $i_{\overline{\operatorname{\mathbf{d}}}}\colon {\mathfrak{M}}^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}\rightarrow {\mathfrak{M}}_{\operatorname{\mathbf{d}}}$ given by forgetting the Harder-Narasimhan filtration, and the diagram $$\xymatrix{
{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}\ar[d]^{p_{\overline{\operatorname{\mathbf{d}}}}}\ar[r]^{i_{\overline{\operatorname{\mathbf{d}}}}}&{\mathfrak{M}}_{\operatorname{\mathbf{d}}}\ar[dl]^{p_{\operatorname{\mathbf{d}}}}\\
{\mathcal{M}}_{\operatorname{\mathbf{d}}}
}$$ commutes.  In the proposition below, the map $q^{\zeta}_{\mu}$ is as defined in (\[qmudef\]). 

\[gcDT\] Let $\zeta$ be a Bridgeland stability condition.  Then there are isomorphisms $$\label{ngwc}
\operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}}}{\mathcal{IC}}_{{\mathfrak{M}}}(\mathbb{Q})\right)\cong{\mbox{\larger[3]{$\boxtimes$}}}_{\oplus, -\infty\xrightarrow{\mu} \infty}^{\operatorname{tw}} q_{\mu,!}^{\zeta}\operatorname{\mathcal{H}}\left(p_{\mu,!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\mu}^{\zeta}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\mu}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)$$ and $$\label{ngwcnc}
\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}}}{\mathcal{IC}}_{{\mathfrak{M}}}(\mathbb{Q})\right)\cong{\mbox{\larger[3]{$\boxtimes$}}}_{\oplus, \infty\xrightarrow{\mu} -\infty}^{\operatorname{tw}} q_{\mu,*}^{\zeta}\operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\mu}^{\zeta}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\mu}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right).$$

Assume in addition that $\zeta$ is generic.  Then there are isomorphisms $$\label{gwPBW}
\operatorname{\mathcal{H}}\left(p_!{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}}}{\mathcal{IC}}_{{\mathfrak{M}}}(\mathbb{Q})\right)\cong {\mbox{\larger[3]{$\boxtimes$}}}_{\oplus,-\infty \xrightarrow{\mu}\infty}^{\operatorname{tw}} \left(q_{\mu,!}^{\zeta}\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})^{\vee}_{\operatorname{vir}}\right)\right)$$ and $$\label{gwPBWnc}
\operatorname{\mathcal{H}}\left(p_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}}}{\mathcal{IC}}_{{\mathfrak{M}}}(\mathbb{Q})\right)\cong {\mbox{\larger[3]{$\boxtimes$}}}_{\oplus, \infty\xrightarrow{\mu} -\infty }^{\operatorname{tw}} \left(q_{\mu,*}^{\zeta}\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right).$$

As in the proof of Theorem \[weakPBW\] we only need to prove the case for which $W=0$ and then we can deduce the general case from the fact that monodromic vanishing cycle functors are exact, and commute with proper maps and the relevant symmetric monoidal structures.  We first prove that the isomorphism (\[ngwc\]) exists.  Fix a $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$. Then if we complete $\geq'$ to a total ordering $\geq$ of $\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$ and define, for each $\overline{\operatorname{\mathbf{d}}}\in\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$, $$\begin{aligned}
{\mathfrak{M}}_{\leq \overline{\operatorname{\mathbf{d}}}}^{\zeta}:=&\bigcup_{\overline{\operatorname{\mathbf{e}}}\leq \overline{\operatorname{\mathbf{d}}}}{\mathfrak{M}}_{\overline{\operatorname{\mathbf{e}}}}^{\zeta}\\
{\mathfrak{M}}_{< \overline{\operatorname{\mathbf{d}}}}^{\zeta}:=&\bigcup_{\overline{\operatorname{\mathbf{e}}}< \overline{\operatorname{\mathbf{d}}}}{\mathfrak{M}}_{\overline{\operatorname{\mathbf{e}}}}^{\zeta}\end{aligned}$$ then ${\mathfrak{M}}_{\leq \overline{\operatorname{\mathbf{d}}}}^{\zeta}\subset {\mathfrak{M}}_{\operatorname{\mathbf{d}}}$ is a closed embedding, and ${\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}\subset{\mathfrak{M}}_{\leq \overline{\operatorname{\mathbf{d}}}}^{\zeta}$ is an open embedding with complement ${\mathfrak{M}}_{<\overline{\operatorname{\mathbf{d}}}}^{\zeta}\subset{\mathfrak{M}}_{\leq \overline{\operatorname{\mathbf{d}}}}^{\zeta}$. We denote by $$\begin{aligned}
i_{\overline{\operatorname{\mathbf{d}}}}\colon &{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}\hookrightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}} \label{id}\\
i_{<\overline{\operatorname{\mathbf{d}}}}\colon &{\mathfrak{M}}_{<\overline{\operatorname{\mathbf{d}}}}^{\zeta}\hookrightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}}\label{idd}\\
i_{\leq\overline{\operatorname{\mathbf{d}}}}\colon &{\mathfrak{M}}_{\leq \overline{\operatorname{\mathbf{d}}}}^{\zeta}\hookrightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}}\label{iddd}\end{aligned}$$ the obvious inclusions.  We will show that all of the terms in the following distinguished triangle are pure, and that the connecting maps are zero, and furthermore that the triangle is split: $$\label{HNtri}
\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},!}i_{\overline{\operatorname{\mathbf{d}}},!}{\mathbb{Q}}_{{\mathfrak{M}}^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}}\right)\rightarrow \operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},!}i_{\leq \overline{\operatorname{\mathbf{d}}},!}{\mathbb{Q}}_{{\mathfrak{M}}^{\zeta}_{\leq \overline{\operatorname{\mathbf{d}}}}}\right)\rightarrow \operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},!}i_{< \overline{\operatorname{\mathbf{d}}},!}{\mathbb{Q}}_{{\mathfrak{M}}^{\zeta}_{< \overline{\operatorname{\mathbf{d}}}}}\right)\rightarrow$$ Fix $\overline{\operatorname{\mathbf{d}}}\in\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$. Consider the following commutative diagram $$\xymatrix{
&{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}^{\zeta}\ar[d]^{q_{\overline{\operatorname{\mathbf{d}}}}}\ar[r]^-{i_{\overline{\operatorname{\mathbf{d}}}}}&{\mathfrak{M}}_{\operatorname{\mathbf{d}}}\ar[ddd]^{p_{\operatorname{\mathbf{d}}}}\\
X^{\zeta}_{\overline{\operatorname{\mathbf{d}}}}/(G_{\operatorname{\mathbf{d}}^1}\times\ldots \times G_{\operatorname{\mathbf{d}}^s})\ar[r]^{q_{2,\overline{\operatorname{\mathbf{d}}}}}\ar[ur]^{q_{1,\overline{\operatorname{\mathbf{d}}}}}&{\mathfrak{M}}_{\operatorname{\mathbf{d}}^1}^{\zeta\operatorname{-ss}}\times\ldots\times{\mathfrak{M}}_{\operatorname{\mathbf{d}}^s}^{\zeta\operatorname{-ss}}\ar[d]^{p^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times p^{\zeta}_{\operatorname{\mathbf{d}}^s}}\\
&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}^1}\times\ldots\times{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}^s}\ar[d]^{q^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times q^{\zeta}_{\operatorname{\mathbf{d}}^s}}\\
&{\mathcal{M}}_{\operatorname{\mathbf{d}}^1}\times\ldots\times{\mathcal{M}}_{\operatorname{\mathbf{d}}^s}\ar[r]^-{\oplus}&{\mathcal{M}}_{\operatorname{\mathbf{d}}}
}$$ where $q_{1,\overline{\operatorname{\mathbf{d}}}}$ is an affine fibration of relative dimension $$f_1(\overline{\operatorname{\mathbf{d}}}):=\sum_{1\leq r<r'\leq s}\operatorname{\mathbf{d}}^r\cdot \operatorname{\mathbf{d}}^{r'}$$ and $q_{2,\overline{\operatorname{\mathbf{d}}}}$ is an affine fibration of relative dimension $$f_2(\overline{\operatorname{\mathbf{d}}}):=\sum_{1\leq r<r'\leq s}\sum_{a\in Q_1}\operatorname{\mathbf{d}}^{r'}_{s(a)}\operatorname{\mathbf{d}}^{r}_{t(a)}.$$ We define $(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}}):=f_1(\overline{\operatorname{\mathbf{d}}})-f_2(\overline{\operatorname{\mathbf{d}}})$. Then by homotopy invariance and the decomposition theorem $$\begin{aligned}
\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},!}i_{\overline{\operatorname{\mathbf{d}}},!}{\mathbb{Q}}_{{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}}\right)
\cong& \operatorname{\mathcal{H}}\left(\oplus_!(q^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times q^{\zeta}_{\operatorname{\mathbf{d}}^s})_!(p^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times p^{\zeta}_{\operatorname{\mathbf{d}}^s})_!{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^1}^{\zeta\operatorname{-ss}}\times\ldots\times{\mathfrak{M}}_{\operatorname{\mathbf{d}}^s}^{\zeta\operatorname{-ss}}}\right)\otimes{\mathbb{L}}^{-(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}})}\\
\cong& \lim_{{\mathbf{f}}\mapsto\infty} \operatorname{\mathcal{H}}\left(\oplus_!(q^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times q^{\zeta}_{\operatorname{\mathbf{d}}^s})_!(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}^1}\times\ldots\times \pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}^s})_!{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}^1}^{\zeta}\times\ldots\times{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}^s}}\right)\otimes{\mathbb{L}}^{-(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}})}\\
\cong&  \oplus_!(q^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times q^{\zeta}_{\operatorname{\mathbf{d}}^s})_!\lim_{{\mathbf{f}}\mapsto\infty}\operatorname{\mathcal{H}}\left((\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}^1}\times\ldots\times \pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}^s})_!{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}^1}^{\zeta}\times\ldots\times{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}^s}}\right)\otimes{\mathbb{L}}^{-(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}})}\\
\cong&\oplus_!(q^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times q^{\zeta}_{\operatorname{\mathbf{d}}^s})_!\operatorname{\mathcal{H}}\left((p^{\zeta}_{\operatorname{\mathbf{d}}^1}\times\ldots\times p^{\zeta}_{\operatorname{\mathbf{d}}^s})_!{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^1}^{\zeta\operatorname{-ss}}\times\ldots\times{\mathfrak{M}}_{\operatorname{\mathbf{d}}^s}^{\zeta\operatorname{-ss}}}\right)\otimes{\mathbb{L}}^{-(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}})}\\
\cong&q^{\zeta}_{\operatorname{\mathbf{d}}^1,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^1,!}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^1}^{\zeta\operatorname{-ss}}}\right)\boxtimes_{\oplus}\ldots\boxtimes_{\oplus} q^{\zeta}_{\operatorname{\mathbf{d}}^s,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^s,!}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^s}^{\zeta\operatorname{-ss}}}\right)\otimes{\mathbb{L}}^{-(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}})}.\end{aligned}$$ Since each $q_{\operatorname{\mathbf{d}}^r}^{\zeta}$ is a proper map, each $q^{\zeta}_{\operatorname{\mathbf{d}}^r,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^r,!}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^r}^{\zeta\operatorname{-ss}}}\right)$ is pure, since as we saw in the proof of Theorem \[weakPBW\], $\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^r,!}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^r}^{\zeta\operatorname{-ss}}}\right)$ is pure.  The claim regarding the distinguished triangles (\[HNtri\]) then follows by induction, and semisimplicity of the category of pure mixed Hodge modules.  Taking care of the twists, we calculate $$\begin{aligned}
\label{int1}
\operatorname{\mathcal{H}}\left(p_!{\mathcal{IC}}_{{\mathfrak{M}}}({\mathbb{Q}})\right)\cong&\operatorname{\mathcal{H}}\left(p_!{\mathbb{Q}}_{{\mathfrak{M}}}\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\\
\nonumber\cong&\bigoplus_{\overline{\operatorname{\mathbf{d}}}\in\operatorname{HN}^{\geq}}\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},!}i_{\overline{\operatorname{\mathbf{d}}},!}{\mathbb{Q}}_{{\mathfrak{M}}_{\overline{\operatorname{\mathbf{d}}}}}\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\\
\nonumber\cong&\bigoplus_{(\operatorname{\mathbf{d}}^1,\ldots,\operatorname{\mathbf{d}}^s)\in\operatorname{HN}^{\geq}}q^{\zeta}_{\operatorname{\mathbf{d}}^1,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^1,!}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^1}^{\zeta\operatorname{-ss}}}\right)\boxtimes_{\oplus}\ldots\boxtimes_{\oplus} q^{\zeta}_{\operatorname{\mathbf{d}}^s,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^s,!}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^s}^{\zeta\operatorname{-ss}}}\right)\otimes{\mathbb{L}}^{-(\overline{\operatorname{\mathbf{d}}},\overline{\operatorname{\mathbf{d}}})+(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\\
\nonumber\cong&\bigoplus_{(\operatorname{\mathbf{d}}^1,\ldots,\operatorname{\mathbf{d}}^s)\in\operatorname{HN}^{\geq}}q^{\zeta}_{\operatorname{\mathbf{d}}^s,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^s,!}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^s}^{\zeta\operatorname{-ss}}}({\mathbb{Q}})\right)\boxtimes^{\operatorname{tw}}_{\oplus}\ldots\boxtimes^{\operatorname{tw}}_{\oplus} q^{\zeta}_{\operatorname{\mathbf{d}}^1,!}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}^1,!}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}^1}^{\zeta\operatorname{-ss}}}({\mathbb{Q}})\right)\\
\nonumber \cong&{\mbox{\larger[3]{$\boxtimes$}}}_{\oplus, -\infty\xrightarrow{\mu} \infty}^{\operatorname{tw}} q_{\mu,!}^{\zeta}\operatorname{\mathcal{H}}\left(p_{\mu,!}^{\zeta}{\mathcal{IC}}_{{\mathfrak{M}}_{\mu}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right).\end{aligned}$$ This completes the proof of the first part of the theorem. 

If $\zeta$ is generic, we may apply Theorem \[weakPBW\] at each slope $\mu\in(-\infty,\infty)$, to deduce (\[gwPBW\]).  The isomorphisms (\[ngwcnc\]) and (\[gwPBWnc\]) are given by taking the Verdier duals of (\[ngwc\]) and (\[gwPBW\]) respectively, by Remark \[twDual\]. 

In the case $W=0$, taking the direct image of the isomorphism (\[ngwcnc\]) of Theorem \[gcDT\] along the map ${\mathcal{M}}(Q)\rightarrow \mathbb{N}^{Q_0}$ we obtain the isomorphism $$\operatorname{H}\left({\mathfrak{M}}(Q),\mathbb{Q}\right)\cong{\mbox{\larger[3]{$\boxtimes$}}}_{+, \infty\xrightarrow{\mu} -\infty}^{\operatorname{tw}} \operatorname{H}\left({\mathfrak{M}}_{\mu}^{\zeta\operatorname{-ss}}(Q),\mathbb{Q}\right).$$ The existence of such an isomorphism is proved by Franzen and Reineke in [@FrRe15] via a vanishing result for even cohomology, and before that for the case of a Dynkin quiver which is not an orientation of $E_8$ by Rimanyi in [@Ri13], where this result is a corollary of the existence of a Poincaré–Birkhoff–Witt isomorphism for hypercohomology. 

Theorems \[weakPBW\] and \[gcDT\] give categorified upgrades of the integrality theorem, and the wall crossing formula, respectively; i.e. we replace these theorems, considered as equalities in a Grothendieck group, with isomorphisms in the appropriate categories.  The rest of the paper is devoted to proving that the isomorphisms in these theorems can be taken to be induced by multiplication in appropriately defined cohomological Hall algebras. 

Cohomological Hall algebras and modules {#CoHAsec}
=======================================

A relative Cohomological Hall algebra {#relc}
-------------------------------------

In this section we define the relative cohomological Hall algebra $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$. Here $\zeta\in\mathbb{H}_+^{Q_0}$ is a stability condition, that we do not assume to be generic.  The underlying cohomologically graded monodromic mixed Hodge module of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ is $\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)$ from Definition \[abbrev\] and Theorem \[weakPBW\].  We will define morphisms $$\label{multint}
\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\right)\boxtimes^{\operatorname{tw}}_{\oplus} \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}\right)\xrightarrow{\operatorname{\mathcal{H}}\left(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}\right)}\\   \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},*} {\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta}\right)$$ for all $\operatorname{\mathbf{d}}=\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''$ with $\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}$. The morphisms (\[multint\]) will satisfy the natural associativity condition for a monoid in the category ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}))$ with the twisted monoidal product $\boxtimes^{\operatorname{tw}}_{\oplus}$. The result is a *relative* version of the cohomological Hall algebra of Kontsevich and Soibelman [@KS2] – relative in the sense that we work over the base ${\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$, instead of the monoid $\Lambda_{\mu}^{\zeta}$ of dimension vectors of slope $\mu$ with respect to the stability condition $\zeta$, as explained in the introduction. 

We define (\[multint\]) as the composition of two morphisms, defined in terms of the commutative diagram $$\xymatrix{
&X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}/\left(G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\right)\ar[dl]_{r_1}\ar[dr]^{r_2}\\
\left(X_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times X_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\right)/\left(G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\right)\ar[ddr]^{p_{\operatorname{\mathbf{d}}'}^{\zeta}\times p_{\operatorname{\mathbf{d}}''}^{\zeta}}\ar[d]^{\cong} &&X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[d]^{\cong}\ar[ddl]_{p_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}}\\
{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times{\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\ar[dr]&&{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\ar[dl]\\
&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar[r]^-{\oplus}&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}. 
}$$ We consider the following composition of isomorphisms $$\begin{aligned}
&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\boxtimes \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\cong^{{\mathtt{TS}}}\\
&\operatorname{\mathcal{H}}\left((p^{\zeta}_{\operatorname{\mathbf{d}}'}\times p^{\zeta}_{\operatorname{\mathbf{d}}''})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}\boxplus{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\cong\\
&\operatorname{\mathcal{H}}\left(((p^{\zeta}_{\operatorname{\mathbf{d}}'}\times p^{\zeta}_{\operatorname{\mathbf{d}}''})\circ r_{1})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}\boxplus{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}''}\circ r_1}}{\mathcal{IC}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}/\left(G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\right)}(\mathbb{Q})\right)\otimes {\mathbb{L}}^{\sum_{a\in Q_1}\operatorname{\mathbf{d}}''_{s(a)}\operatorname{\mathbf{d}}'_{t(a)}/2}\cong\\
&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\end{aligned}$$ to obtain the isomorphism $$\begin{aligned}
\alpha^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\colon&\oplus_*\left(\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\boxtimes \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\right)\xrightarrow{\cong}\\& \oplus_*\left(\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\right).\end{aligned}$$ Next consider the commutative diagram of stacks $$\label{smap}
\xymatrix{
{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[d]^{\cong}\ar[rr]^-{s^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}&&{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\ar[d]^{\cong}\\
X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[d]^{p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\ar[r]^{\iota^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}&X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[r]^{r^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}&X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}\ar[d]^{p^{\zeta}_{\operatorname{\mathbf{d}}}}\\
{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar[rr]^{\oplus}&&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}. 
}$$ Note that $\iota_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^\zeta$ is a closed inclusion, while $r_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^\zeta$ is proper and representable, and so it follows that $s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^\zeta$ is a proper and representable morphism of stacks. 

We define $V_N=\prod_{i\in Q_0}\operatorname{Hom}(\mathbb{C}^N,\mathbb{C}^{\operatorname{\mathbf{d}}_i})$, and $U_N=\prod_{i\in Q_0}\operatorname{Hom}^{\mathrm{surj}}(\mathbb{C}^N,\mathbb{C}^{\operatorname{\mathbf{d}}_i})$. Then for sufficiently large $N$, $G_{\operatorname{\mathbf{d}}}$ acts freely on $U_N$, as does $G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\subset G_{\operatorname{\mathbf{d}}}$. We define $$\begin{aligned}
\label{XNdef}&X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}},N}=X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}\times_{G_{\operatorname{\mathbf{d}}}}U_N\\
\label{XXNdef}&X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}=X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\times_{G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}U_{N},\end{aligned}$$ and let ${\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}},N}$ and ${\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}$ denote the functions induced by ${\mathrm{Tr}(W)}$ on $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}},N}$ and $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}$ respectively.  Passing to the limit of the composition of morphisms $$\begin{aligned}
\label{preLim}
\beta^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}\colon &\oplus_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}\right)\xrightarrow{\cong }
\operatorname{\mathcal{H}}\left((p^{\zeta}_{\operatorname{\mathbf{d}},N}\circ s^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N})_*{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}\right)\xrightarrow{\cong}\\ \nonumber
&\rightarrow \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}},N}}}s^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\mathbb{Q}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right)\rightarrow \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}},N}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}},N}^{\zeta\operatorname{-ss}}}\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')}\end{aligned}$$ we obtain $$\begin{aligned}
\label{betadef}
\beta^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\colon &\oplus_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\rightarrow \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}({\mathbb{Q}})\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}.\end{aligned}$$ Here we have used that $\oplus_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)$ is naturally isomorphic to $\operatorname{\mathcal{H}}\left(\oplus_*p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)$ since $\oplus_*$ is exact, as $\oplus$ is finite.  Composing the appropriate twists of $\beta^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ and $\alpha^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ gives the desired morphism $$\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}):= \left(\beta^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\otimes {\mathbb{L}}^{-(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\right)\circ \left(\alpha^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\otimes {\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2}\right)$$ of monodromic mixed Hodge modules.  We define $$\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}^{\zeta}_{W,\mu})=\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}}\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}).$$ The proof that the resulting structure is associative is standard, and is as in [@KS2]. 

We denote by $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$ the monoid $\left(\operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right),\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}^{\zeta}_{W,\mu}),{\mathcal{IC}}_{{\mathcal{M}}_0^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)$ in $\left({\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})),\boxtimes^{\operatorname{tw}}_{\oplus}\right)$.

Recall from Proposition \[cvs\] the isomorphisms $\nu_{\mu}\colon{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\mu}}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{0,\mu}^{\zeta}\right)\cong \operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)$. The following technical lemma will be used in proving the strong versions of the PBW theorem, as stated in Theorems \[qea\] and \[strongPBW\]. 

\[cohacom\] The following diagram commutes: $$\xymatrix{
{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\left(\operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{0,\mu}^{\zeta}\right)\boxtimes_{\oplus}^{\operatorname{tw}} \operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{0,\mu}^{\zeta}\right)\right)\ar[rr]^-{{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\mu}}}\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}^{\zeta}_{0,\mu})}\ar[d]^{(\nu_{\mu}\boxtimes^{\operatorname{tw}}_{\oplus} \nu_{\mu})\circ {\mathtt{TS}}^{-1}} &&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\mu}}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{0,\mu}^{\zeta}\right)\ar[d]^{\nu}\\
\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\boxtimes^{\operatorname{tw}}_{\oplus} \operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\ar[rr]^-{\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}^{\zeta}_{W,\mu})}&& \operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right). 
}$$

Here ${\mathtt{TS}}$ is the Thom–Sebastiani isomorphism.  We break the two horizontal arrows into their constituent parts, given by the constituent morphisms of the composition $\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}^{\zeta}_{W,\mu})$. Then the problem reduces to several trivial commutativity statements regarding smaller squares. 

Taking the direct image along $\dim \colon{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}\rightarrow\mathbb{N}^{Q_0}$ we obtain an element $\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\right)$ in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ that is noncanonically isomorphic to the underlying $\mathbb{N}^{Q_0}$-graded monodromic mixed Hodge module of the absolute cohomological Hall algebra $\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$, a monoid in the monoidal category $({\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0})),\boxtimes^{\operatorname{tw}}_+)$, defined below.  I.e. we obtain the monoid in $({\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}),\boxtimes^{\operatorname{tw}}_{+})$ $$\label{premdef}
\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})):=\left(\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta})\right),\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})\right), \dim_*{\mathcal{IC}}_{0}(\mathbb{Q})\right).$$ The notation in the left hand side of (\[premdef\]) will become more transparent after we introduce the perverse filtration in Section \[PervSec\]. 

We complete the preceding remark by recalling the definition of the cohomological Hall algebra $\operatorname{H}(\mathcal{A}^{\zeta}_{W,\mu})$ from [@KS2].  Firstly, mimicking the construction of $\alpha^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$, there is an isomorphism $$\begin{aligned}
\overline{\alpha}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\colon &\operatorname{\mathcal{H}}\left((\dim\circ p_{\operatorname{\mathbf{d}}'}^{\zeta})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}}(\mathbb{Q})\right)\boxtimes_{+}\operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}''})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\rightarrow \\&\operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}.\end{aligned}$$ Similarly, with notation as in (\[preLim\]), applying $(\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}},N})_*{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}_{\operatorname{\mathbf{d}},N}^{\zeta}}}$ to the Verdier dual of $${\mathbb{Q}}_{X_{\operatorname{\mathbf{d}},N}^{\zeta\operatorname{-ss}}}\rightarrow s^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}$$ and passing to the limit, we obtain the morphism $$\overline{\beta}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\colon\operatorname{\mathcal{H}}\left((\dim\circ p_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\rightarrow \operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}.$$ Composing, we obtain the multiplication for the cohomological Hall algebra $(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}),\operatorname{H}(\ast_{W,\mu}^{\zeta}),\mathbb{Q}_{\{0\}})$ defined in [@KS2], where $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}):=\operatorname{\mathcal{H}}\left((\dim\circ p_{\mu}^{\zeta})_*{\mathfrak{IC}}^{\zeta}_{W,\mu}\right)$ and $\operatorname{H}(\ast_{W,\mu}^{\zeta}):=\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}}\operatorname{H}(\ast_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})$, with $$\operatorname{H}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}):=\left(\overline{\beta}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\otimes {\mathbb{L}}^{-(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\right)\circ \left(\overline{\alpha}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\otimes {\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2}\right).$$

Perverse filtration {#PervSec}
-------------------

Since ${\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ is smooth, and $\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\colon {\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\rightarrow {\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$ is proper, by the decomposition theorem there is an isomorphism $$\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\cong\operatorname{\mathcal{H}}\left(\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right).$$ Although this decomposition is not canonical, the result implies that the natural maps $\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\rightarrow \pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}$ admit left inverses, where $\tau_{\leq p}$ is the truncation functor with respect to the usual t structure on the category of monodromic mixed Hodge modules.  In particular, the maps $$\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)\rightarrow \operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)$$ are inclusions, inducing the *perverse filtration* on $\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)$, with $$\operatorname{\mathfrak{P}}^p\left(\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)\right):=\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right).$$ For $W\neq 0$, we consider the split inclusion $$\begin{aligned}
&\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\xrightarrow{\cong}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\hookrightarrow {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\xrightarrow{\cong}\\ &\rightarrow\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\end{aligned}$$ where the first and last maps are isomorphisms by properness of $\pi_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ and exactness of ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}$, and the middle map is a split inclusion as it is obtained by applying the functor ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}$ to a split inclusion.  We set $$\operatorname{\mathfrak{P}}^p\left(\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)\right):=\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right).$$ The perverse filtration is well-defined in the limit, as for fixed $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$, fixed cohomological degree $n$, and for sufficiently large ${\mathbf{f}}'>{\mathbf{f}}$, in the commutative diagram $$\xymatrix{
\operatorname{H}^n\left({\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta},\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-\dim({\mathfrak{M}}_{\operatorname{\mathbf{d}}})/2}\right)\ar[r] &\operatorname{H}^n\left({\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-\dim({\mathfrak{M}}_{\operatorname{\mathbf{d}}})/2}\right)\\
\operatorname{H}^n\left({\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta},\tau_{\leq p}\left(\pi ^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-\dim({\mathfrak{M}}_{\operatorname{\mathbf{d}}})/2}\right)\right)\ar[r]\ar[u]& \operatorname{H}^n\left({\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},\tau_{\leq p}\left(\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\mathbb{Q}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-\dim({\mathfrak{M}}_{\operatorname{\mathbf{d}}})/2}\right)\right)\ar[u],
}$$ the horizontal maps are isomorphisms by Lemma \[appLem\].  Since the constituent morphisms of $\operatorname{H}(\ast_{W,\mu}^{\zeta})$ lift to morphisms of monodromic mixed Hodge modules on ${\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}$, it follows that $\operatorname{H}(\ast_{W,\mu}^{\zeta})$ respects the perverse filtration, and from the definitions, along with the decomposition theorem, we obtain the following proposition. 

\[PCoha\] There is a natural isomorphism of monoids in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ $$\operatorname{Gr}_{\operatorname{\mathfrak{P}}}\left(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}),\operatorname{H}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta}),\operatorname{H}(\operatorname{\mathcal{A}}_{W,0}^{\zeta})\right)\cong\operatorname{\mathcal{H}}\left(\dim_* \left(\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta}),\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}^{\zeta}_{W,\mu}),{\mathcal{IC}}_{{\mathcal{M}}_0}^{\zeta\operatorname{-ss}}(\mathbb{Q})\right)\right).$$ Furthermore, forgetting the algebra structure, there is a noncanonical isomorphism of underlying monodromic mixed Hodge modules $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}))\cong\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$.

The isomorphism of algebras is what justifies the notation of (\[premdef\]). 

The following technical lemma is what will enable us to use the localised coproduct on $\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$ to induce a Hopf algebra structure on the associated graded object $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}))\cong\operatorname{\mathcal{H}}\left(\dim_* \operatorname{\mathcal{H}}(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta})\right)$. It is only a very slight variation of [@deCat12 Prop.1.4.4], but we include the proof for completeness. 

\[HCM\] Let $V$ be a vector bundle on ${\mathfrak{M}}_{\operatorname{\mathbf{d}}}$, and let ${\mathfrak{eu}}(V)\in\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}},\mathbb{Q})$ be the corresponding equivariant Euler class.  Then $${\mathfrak{eu}}(V)\cdot \operatorname{\mathfrak{P}}^p(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},\mathbb{Q}))\subset \operatorname{\mathfrak{P}}^{p+2\dim(V)}(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},\mathbb{Q})).$$

Let $\operatorname{pr}\colon T(V)\rightarrow {\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$ be the projection from the total space of $V$ restricted to ${\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$, and let $i\colon {\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\rightarrow T(V)$ be the inclusion of the zero section.  Let $T_{\mathbf{f}}(V)$ be the total space of the bundle $V$ pulled back along the map ${\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\rightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$ defined by forgetting the framing, and let $\operatorname{pr}_{{\mathbf{f}}}$ and $i_{{\mathbf{f}}}$ denote the corresponding projections and inclusions.  Then we have the equality $$\cdot {\mathfrak{eu}}(V)|_{\operatorname{H}({\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},\mathbb{Q})}=\operatorname{H}\left(i_{{\mathbf{f}},*}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\rightarrow \mathbb{Q}_{T_{{\mathbf{f}}}(V)}\otimes{\mathbb{L}}^{-\dim(V)}\rightarrow  i_{{\mathbf{f}},*}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\otimes{\mathbb{L}}^{-\dim(V)}\right)$$ and the action of $\cdot{\mathfrak{eu}}(V)$ on $\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},\mathbb{Q})$ is given by the morphism $$\operatorname{H}\left(\pi^{\zeta} _{{\mathbf{f}},\operatorname{\mathbf{d}},*}\left(\operatorname{pr}_{{\mathbf{f}},*}i_{{\mathbf{f}},*}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\rightarrow \operatorname{pr}_{{\mathbf{f}},*}i_{{\mathbf{f}},*}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\otimes{\mathbb{L}}^{-\dim(V)}\right)\right)$$ for ${\mathbf{f}}\gg 0$, which respects the perverse filtration on $\operatorname{H}({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\mathbb{Q})$ (with the shift by $2\dim_{\mathbb{C}}(V)$) since $$\pi^{\zeta} _{{\mathbf{f}},\operatorname{\mathbf{d}},*}\left(\operatorname{pr}_{{\mathbf{f}},*}i_{{\mathbf{f}},*}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\rightarrow \operatorname{pr}_{{\mathbf{f}},*}i_{{\mathbf{f}},*}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\otimes{\mathbb{L}}^{-\dim(V)}\right)$$ is a map of mixed Hodge modules on ${\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$. The result then follows from the definition of the perverse filtration on $\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},\mathbb{Q})$.

Relative CoHA modules
---------------------

Let $\zeta\in \mathbb{H}_+^{Q_0}$ be a stability condition.  For each framing vector ${\mathbf{f}}\in\mathbb{N}^{Q_0}$, and each slope $\mu\in(-\infty,\infty)$, we will form a module $$\begin{aligned}
\operatorname{\mathcal{H}}(\mathcal{F}^{\zeta}_{W,{\mathbf{f}},\mu})\in{\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))\end{aligned}$$ for the monoid $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$. Note that for the definition we do not require that $\zeta$ is $\mu$-generic.  Let $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$. We define $$\begin{aligned}
{\mathfrak{IC}}^{\zeta}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}:=&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}(\mathbb{Q})\otimes {\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2}\\
{\mathfrak{IC}}^{\zeta}_{W,{\mathbf{f}},\mu}:=&\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}{\mathfrak{IC}}^{\zeta}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\end{aligned}$$ in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\mu}))$. The Tate twist is chosen so that if $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$ then $$\operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta_{{\mathbf{f}}}^{[\mu]}}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{[\mu]}}_{(1,\operatorname{\mathbf{d}})}}}\mathbb{Q}_{X(Q_{\mathbf{f}})^{\zeta_{{\mathbf{f}}}^{[\mu]}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}/G_{\operatorname{\mathbf{d}}}}\rightarrow p_{\mu,*}^{\zeta_{{\mathbf{f}}}^{[\mu]}}j^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\right),$$ where $j_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ is as in (\[jmap\]), gives rise to an *untwisted* morphism of complexes of monodromic mixed Hodge modules $$\label{shiftjust}
\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}):=\operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}^{\zeta}_{W,\operatorname{\mathbf{d}}}\right)\rightarrow \operatorname{\mathcal{H}}\left(\pi _{{\mathbf{f}},\mu,*}^{\zeta}{\mathfrak{IC}}^{\zeta}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\right).$$ via the natural isomorphism $$\operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta_{{\mathbf{f}}}^{[\mu]}}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{[\mu]}}_{(1,\operatorname{\mathbf{d}})}}}\mathbb{Q}_{X(Q_{\mathbf{f}})^{\zeta_{{\mathbf{f}}}^{[\mu]}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}/G_{\operatorname{\mathbf{d}}}}\right)\otimes {\mathbb{L}}^{{}-\dim({\mathfrak{M}}_{d})/2}\rightarrow \operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}^{\zeta}_{W,\operatorname{\mathbf{d}}}\right)$$ provided by homotopy invariance.  Later we will see (see Theorem \[cyclesprop\]) that (\[shiftjust\]) is in fact a morphism in the category of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$-modules after taking the direct sum over all $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$.

We define $$\begin{aligned}
{\mathfrak{IC}}^{\zeta}_{W,{\mathbf{f}},\mu}:=&\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda^{\zeta}_{\mu}}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},\\
\operatorname{\mathcal{H}}(\mathcal{F}^{\zeta}_{W,{\mathbf{f}},\mu}):=&\operatorname{\mathcal{H}}\left(\pi _{{\mathbf{f}},\mu,*}^{\zeta}{\mathfrak{IC}}^{\zeta}_{W,{\mathbf{f}},\mu}\right).\end{aligned}$$ Let $\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''=\operatorname{\mathbf{d}}$, with $\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}$. Our goal is to define maps $$\begin{aligned}
& \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\right)\boxtimes_{\oplus}^{\operatorname{tw}} \operatorname{\mathcal{H}}\left(\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}\right)\xrightarrow{\operatorname{\mathcal{H}}(\operatorname{\tilde{\cdot}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})}\operatorname{\mathcal{H}}\left(\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\right)\end{aligned}$$ satisfying the obvious associativity constraint with respect to the relative cohomological Hall algebra multiplication $\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})$. Consider the commutative diagram $$\label{modDiag}
\xymatrix{
X(Q_{{\mathbf{f}}})^{(\zeta,\zeta_{{\mathbf{f}}}^{(\mu)})\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}\ar[d]_{r_1}/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\ar[r]^-{r_2}&X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{(\zeta,\zeta_{{\mathbf{f}}}^{(\mu)})\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[d]_{p^{\zeta,\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}
\\
\left( X(Q)_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times Y_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}\right)/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\ar[ddr]_-{p^{\zeta}_{\operatorname{\mathbf{d}}'}\times \pi _{{\mathbf{f}},\operatorname{\mathbf{d}}''}}\ar[d]^{\cong}&  {\mathcal{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}\ar[dd]_{\operatorname{id}\times\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}} & X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar@{_(->}[ul]^h\ar[ddl]^{\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}\ar[l]^-{p^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}
\\
{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''} &   & 
\\
&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}. 
}$$ The space $Y_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}$ in the above diagram is the scheme of stable framed representations defined at the start of Section \[cohaDT\].  The map $h$ is defined to be the natural inclusion, and $\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}$ is defined to be the composition making the diagram commute.  In order to elucidate a little what the inclusion $h$ looks like, we describe its complement.  The stack $X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{(\zeta,\zeta_{{\mathbf{f}}}^{(\mu)})\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ is naturally isomorphic to the stack of short exact sequences $$\label{jexp}
0\rightarrow \rho'\rightarrow \rho\rightarrow\rho''\rightarrow 0$$ where $\rho'$ is a $\operatorname{\mathbf{d}}'$-dimensional $\zeta$-semistable $\mathbb{C}Q$-representation and $\rho''$ is a $(1,\operatorname{\mathbf{d}}'')$-dimensional $\zeta_{{\mathbf{f}}}^{(\mu)}$-stable $\mathbb{C}Q_{{\mathbf{f}}}$-representation equipped with a framing $\rho''_{\infty}\cong\mathbb{C}$. The complement to the inclusion $h$ consists of those pairs such that the representation $\rho$ is not itself $\zeta_{{\mathbf{f}}}^{(\mu)}$-stable.  For example if (\[jexp\]) splits and $\operatorname{\mathbf{d}}'\neq 0$, it represents an element of the complement. 

We define $\gamma^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}=(\operatorname{id}\times \pi_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta})\circ p^{\zeta,\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}$. Consider the following composition of morphisms $$\begin{aligned}
&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\boxtimes \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}}(\mathbb{Q})\right)\cong^{{\mathtt{TS}}}\\
&\operatorname{\mathcal{H}}\left((p^{\zeta}_{\operatorname{\mathbf{d}}'}\times\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}\boxplus{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\cong\\
&\operatorname{\mathcal{H}}\left((p^{\zeta}_{\operatorname{\mathbf{d}}'}\times\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''})_*r_{1,*}{\phi^{\operatorname{mon}}_{({\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}\boxplus{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''})\circ r_1}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{(\zeta,\zeta_{{\mathbf{f}}}^{(\mu)})\operatorname{-ss}}/(G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''})}(\mathbb{Q})\right)\otimes {\mathbb{L}}^{\sum_{a\in Q_1}\operatorname{\mathbf{d}}''_{s(a)}\operatorname{\mathbf{d}}'_{t(a)}/2}\cong\\
&\operatorname{\mathcal{H}}\left(\gamma^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta,\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})^{(\zeta,\zeta_{{\mathbf{f}}}^{(\mu)})\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes {\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\rightarrow\\
&\operatorname{\mathcal{H}}\left(\gamma^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}h_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\cong\\
&\operatorname{\mathcal{H}}\left(\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\end{aligned}$$ defining the map $$\begin{aligned}
\alpha^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\colon&\oplus_*\left( \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\boxtimes \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}}(\mathbb{Q})\right)\right)\rightarrow\\
&\oplus_*\left(\operatorname{\mathcal{H}}\left(\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}.\end{aligned}$$ Consider the commutative diagram $$\label{modDiag2}
\xymatrix{
X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar@/^2pc/[rrrr]^-{s_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}\ar[d]^{\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}\ar[rr]_{\iota_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}&&Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[rr]_-{r_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}&&Y^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}\ar[d]^{\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\\
{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar[rrrr]^{\oplus}&&&&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}. 
}$$ We set $U_N=\prod_{i\in Q_0}\operatorname{Hom}^{\mathrm{surj}}(\mathbb{C}^N,\mathbb{C}^{\operatorname{\mathbf{d}}_i})$, and define $$\begin{aligned}
X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}''),N}=&X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}\times_{G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}U_N\\
Y^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N}=&Y^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\times_{G_{\operatorname{\mathbf{d}}}}U_N.\end{aligned}$$ Taking the limit in cohomology as $N\mapsto \infty$ of the composition of morphisms $$\begin{aligned}
&\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}^{\zeta}s_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N,*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}''),N}}}{\mathbb{Q}}_{X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}''),N}}  \xrightarrow{\cong}\\\rightarrow &\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{(1,\operatorname{\mathbf{d}}),N}}}s_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N,*}{\mathbb{Q}}_{X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}''),N}}\rightarrow \pi_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{(1,\operatorname{\mathbf{d}}),N}}}{\mathbb{Q}}_{Y^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N}}\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\end{aligned}$$ we obtain the composition $\beta^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}:$ $$\begin{aligned}
&\oplus_*\operatorname{\mathcal{H}}\left(\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\cong\\
&\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}s_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\rightarrow\\
&\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}.\end{aligned}$$ We define $$\operatorname{\mathcal{H}}(\operatorname{\tilde{\cdot}}_{W,f,\mu}^{\zeta}):= \left(\beta^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\otimes {\mathbb{L}}^{-(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2+{\mathbf{f}}\cdot \operatorname{\mathbf{d}}''/2}\right)\circ \left(\alpha^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\otimes {\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2+{\mathbf{f}}\cdot \operatorname{\mathbf{d}}''/2}\right).$$ As in the case of the cohomological Hall algebra, there is an analogous construction in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\mathbb{N}^{Q_0}))$ making $$\operatorname{H}(\operatorname{\mathcal{F}}_{W,{\mathbf{f}},\mu}^{\zeta}):=\operatorname{\mathcal{H}}\left((\dim\circ \pi^{\zeta}_{\mu})_*{\mathfrak{IC}}_{W,{\mathbf{f}},\mu}\right)$$ into a module over $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$. In brief, we define a map $$\begin{aligned}
\overline{\alpha}^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\colon &+_*\left( \operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}'})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\boxtimes \operatorname{\mathcal{H}}\left((\dim\circ \pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''})_*{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\right)\rightarrow\\
&+_*\left(\operatorname{\mathcal{H}}\left((\dim\circ \tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\end{aligned}$$ in the same way as $\alpha^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}$, and a map $$\begin{aligned}
\overline{\beta}^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\colon &+_*\operatorname{\mathcal{H}}\left((\dim\circ \tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\rightarrow\\
&\operatorname{\mathcal{H}}\left((\dim\circ \pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}})_*{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{Y^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\end{aligned}$$ in the same way as $\beta^{\zeta}_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}$, and composing appropriate twists of these maps, we obtain the desired map $$\operatorname{H}(\operatorname{\tilde{\cdot}}_{W,{\mathbf{f}},\mu}^{\zeta})\colon \operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})\boxtimes_+^{\operatorname{tw}}\operatorname{H}(\operatorname{\mathcal{F}}_{W,{\mathbf{f}},\mu}^{\zeta})\rightarrow \operatorname{H}(\operatorname{\mathcal{F}}_{W,{\mathbf{f}},\mu}^{\zeta}).$$ Since $\dim\circ \pi^{\zeta}_{{\mathbf{f}},\mu}$ factors through the proper map $\pi^{\zeta}_{{\mathbf{f}},\mu}$, we obtain a perverse filtration $\operatorname{\mathfrak{P}}$ on $\operatorname{H}(\operatorname{\mathcal{F}}_{W,{\mathbf{f}},\mu}^{\zeta})$ via the construction at the start of Section \[PervSec\], and the following companion to Proposition \[PCoha\]. 

\[PCom\] The module structure $\operatorname{H}(\operatorname{\tilde{\cdot}}_{W,\mu}^{\zeta})$ induces a module structure $$\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\tilde{\cdot}}_{W,\mu}^{\zeta}))\colon \operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}))\boxtimes_+^{\operatorname{tw}}\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{F}}^\zeta_{W,{\mathbf{f}},\mu}))\rightarrow \operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{F}}^\zeta_{W,{\mathbf{f}},\mu}))$$ and there is an isomorphism of $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^\zeta))$-modules $$\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{F}}^{\zeta}_{W,{\mathbf{f}},\mu}))\cong\operatorname{\mathcal{H}}(\dim_*\operatorname{\mathcal{H}}(\operatorname{\mathcal{F}}^{\zeta}_{W,{\mathbf{f}},\mu})).$$

Proof of Theorem \[repthm\] {#rep_thm_proof}
---------------------------

Recall that we define $\zeta_{{\mathbf{f}}}^{[\mu]}$ to be the stability condition in $\mathbb{H}_+^{(Q_{{\mathbf{f}}})_0}$ obtained by setting the slope of $S_{\infty}$, the simple module concentrated at vertex $\infty$, to be $\mu$, and setting $\lvert Z(S_{\infty})\rvert=1$. A $\mathbb{C}Q_{{\mathbf{f}}}$-module of dimension $(1,\operatorname{\mathbf{d}})$, where $\operatorname{\mathbf{d}}$ has slope $\mu$, is $\zeta_{{\mathbf{f}}}^{[\mu]}$-semistable if and only if the underlying $\mathbb{C}Q$-module is $\zeta$-semistable.  A $\zeta_{{\mathbf{f}}}^{[\mu]}$-semistable $\mathbb{C}Q_{{\mathbf{f}}}$-module $N$ of dimension vector $(1,\operatorname{\mathbf{d}})$ admits a unique Harder-Narasimhan filtration with respect to the stability condition $\zeta_{{\mathbf{f}}}^{(\mu)}$, which is either the trivial filtration of $N$, or is of the form $0\subset N'\subset N$, where $N'$ is a $\zeta_{{\mathbf{f}}}^{(\mu)}$-stable $\mathbb{C}Q_{{\mathbf{f}}}$-module of dimension vector $(1,\operatorname{\mathbf{d}}')$ for $\operatorname{\mathbf{d}}'<\operatorname{\mathbf{d}}$, and $N/N'$ has dimension vector zero, when restricted to the framing vertex $\infty$, and is $\zeta$-semistable when considered as a $\mathbb{C}Q$-module.  Geometrically, this is expressed in the following proposition, which is a special case of [@Reineke_HN Prop.3.4]. 

There is a stratification $$X^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}={\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\coprod \left(\coprod_{\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''=\operatorname{\mathbf{d}}}X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\right)$$ with $j\colon {\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\hookrightarrow X^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}$ the inclusion of the open stratum, if the domain of $j$ is nonempty. 

Denote by $\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\colon X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}\rightarrow{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$ the composition of $p^{\zeta}_{\operatorname{\mathbf{d}}}$ with the projection of the affine fibration $X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}\rightarrow {\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$.

\[splitinj\] The map $\operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}j_!\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\right)\rightarrow \operatorname{\mathcal{H}}\left(\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}\mathbb{Q}_{X^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}\right)$ is a split injection in each cohomological degree. 

We complete the partial order on $\operatorname{HN}^{\geq}_{(1,\operatorname{\mathbf{d}})}$ to a total order $<$ as in the proof of Theorem \[gcDT\].  We define a total order on dimension vectors $\operatorname{\mathbf{d}}'\leq \operatorname{\mathbf{d}}$ by setting $\operatorname{\mathbf{d}}'\leq_{\operatorname{t}} \operatorname{\mathbf{e}}'$ if $((1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}-\operatorname{\mathbf{d}}')<((1,\operatorname{\mathbf{e}}'),\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}')$, or if $\operatorname{\mathbf{e}}'=\operatorname{\mathbf{d}}$. For each $\operatorname{\mathbf{d}}'\leq_t \operatorname{\mathbf{d}}$, let $$i_{\operatorname{\mathbf{d}}'}\colon X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\rightarrow X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}$$ be the embedding.  We define the embeddings $i_{<_{\operatorname{t}}\operatorname{\mathbf{d}}'}$ and $i_{\leq_{\operatorname{t}} \operatorname{\mathbf{d}}'}$ similarly (see (\[id\]),(\[idd\]),(\[iddd\])).  Then as in the proof of Theorem \[gcDT\], for each $\operatorname{\mathbf{d}}'\leq_t \operatorname{\mathbf{d}}$ there is a distinguished triangle $$\begin{aligned}
\label{trin}
&\operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}i_{\operatorname{\mathbf{d}}',!}\mathbb{Q}_{X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}-\operatorname{\mathbf{d}}'}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}-\operatorname{\mathbf{d}}'}}\right)\rightarrow \operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}i_{\leq \operatorname{\mathbf{d}}',!}\mathbb{Q}_{\bigcup_{\operatorname{\mathbf{e}}\leq_{\operatorname{t}} \operatorname{\mathbf{d}}'}X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{e}}),\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}/G_{\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}}\right)\\ \nonumber \rightarrow
&\operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}i_{<_{\operatorname{t}}\operatorname{\mathbf{d}}',!}\mathbb{Q}_{\bigcup_{\operatorname{\mathbf{e}}<_{\operatorname{t}}\operatorname{\mathbf{d}}'}X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{e}}),\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}/G_{\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}}\right)\xrightarrow{\varrho_{\operatorname{\mathbf{d}}'}}.\\\end{aligned}$$ We set $\operatorname{\mathbf{d}}''=\operatorname{\mathbf{d}}-\operatorname{\mathbf{d}}'$. Consider the following commutative diagram $$\xymatrix{
{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'}\times{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar[dr]_{\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'}\times p^{\zeta}_{\operatorname{\mathbf{d}}''}}&\ar[l]_-{q}X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[d]^{\Psi}\ar[r]^-{i_{\operatorname{\mathbf{d}}'}}&Y^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}\ar[d]^{\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\\
&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar[r]^-{\oplus}&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}
}$$ where $q$ is the natural map.  Then there are isomorphisms $$\begin{aligned}
\label{purp}
\operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}i_{\operatorname{\mathbf{d}}',!}\mathbb{Q}_{X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\right)\cong &\oplus_*\operatorname{\mathcal{H}}\left(\Psi_!\mathbb{Q}_{X(Q_{{\mathbf{f}}})^{(\zeta_{{\mathbf{f}}}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}'),\operatorname{\mathbf{d}}''}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\right)
\\ \nonumber \cong &\oplus_*\operatorname{\mathcal{H}}\left((\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'}\times p^{\zeta}_{\operatorname{\mathbf{d}}''})_!\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'}\times{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}}\right)\otimes\mathbb{L}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')}
\\ \nonumber \cong &\oplus_*\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}',!}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'}}\boxtimes\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}}'',!}^{\zeta}\mathbb{Q}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}\right)\otimes\mathbb{L}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')}\right)\end{aligned}$$ and the final mixed Hodge module is pure, since $\pi_{{\mathbf{f}},\operatorname{\mathbf{d}}'}^{\zeta}$ and $\oplus$ are proper, and $\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}}'',!}^{\zeta}\mathbb{Q}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}\right)$ is pure by Lemma \[appLem\].  By induction on $\operatorname{\mathbf{d}}'$ it follows that all terms in (\[trin\]) are pure, and so the connecting map $\varrho_{\operatorname{\mathbf{d}}'}$ is zero for all $\operatorname{\mathbf{d}}'$, and in particular $\varrho_{\operatorname{\mathbf{d}}}=0$ and so $$\operatorname{\mathcal{H}}^n\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}j_!\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\right)\rightarrow \operatorname{\mathcal{H}}^n\left(\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}\mathbb{Q}_{X^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}\right)$$ is an injection for all $n$. This injection splits because it belongs to the semisimple category of pure mixed Hodge modules on ${\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$.

Since the restriction of ${\mathcal{T}r(W)}_{\mu}^{\zeta}$ to ${\mathcal{M}}_0^{\zeta\operatorname{-ss}}$ is zero, there are natural isomorphisms $$\begin{aligned}
\operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}},\mu,*}^{\zeta}{\mathfrak{IC}}_{W,{\mathbf{f}},\mu}^\zeta\right)\Big|_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_0}\cong&\operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}},\mu,*}^{\zeta}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},0}^{\zeta}}\right)
\\&\cong{\mathcal{IC}}_{{\mathcal{M}}_0^{\zeta\operatorname{-ss}}}({\mathbb{Q}})
\\&\cong\mathbb{Q}_{{\mathcal{M}}_0}.\end{aligned}$$ Denote by $$\kappa_{{\mathbf{f}}}\colon \mathbb{Q}_{{\mathcal{M}}_0}\rightarrow \operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}},\mu,*}^{\zeta}{\mathfrak{IC}}_{W,{\mathbf{f}},\mu}^\zeta\right)$$ the natural inclusion. 

\[cyclesprop\] Define $\Theta_W$ to be the following composition of maps: $$\begin{aligned}
&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\xrightarrow{\cong} \operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\boxtimes_{\oplus}^{\operatorname{tw}} \mathbb{Q}_{{\mathcal{M}}_0}\xrightarrow{\operatorname{id}\boxtimes^{\operatorname{tw}}_{\oplus} \kappa_{{\mathbf{f}}}}\\
&\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda^{\zeta}_{\mu}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\right)\boxtimes_{\oplus}^{\operatorname{tw}} \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}\right)\xrightarrow{\operatorname{\mathcal{H}}(\operatorname{\tilde{\cdot}}_{W,\mu}^{\zeta})}\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\mu,*}{\mathfrak{IC}}_{W,{\mathbf{f}},\mu}^{\zeta}\right).\end{aligned}$$ Then $\Theta_W$ is a map of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$-modules $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})\rightarrow \operatorname{\mathcal{H}}(\operatorname{\mathcal{F}}_{W,{\mathbf{f}},\mu}^{\zeta})$ that is a split surjection in $\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}})$ in each cohomological degree. 

Putting $W=0$ and fixing $\operatorname{\mathbf{d}}\in\Lambda^{\zeta}_{\mu}$, the Verdier dual of the map $$\Theta_0\colon \operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},*}{\mathfrak{IC}}^{\zeta}_{0,\operatorname{\mathbf{d}}}\right)\rightarrow\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}^{\zeta}_{0,{\mathbf{f}},\operatorname{\mathbf{d}}}\right)$$ is a Tate twist of the map of Lemma \[splitinj\], via the isomorphism $$\operatorname{\mathcal{H}}\left(\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}\mathbb{Q}_{X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}}\right)\cong\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},!}{\mathfrak{IC}}^{\zeta}_{0,\operatorname{\mathbf{d}}}\right)\otimes{\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}$$ provided by homotopy invariance.  In particular, $\operatorname{\mathcal{H}}^n\left(p^{\zeta}_{\operatorname{\mathbf{d}},*}{\mathfrak{IC}}^{\zeta}_{0,\operatorname{\mathbf{d}}}\right)\rightarrow\operatorname{\mathcal{H}}^n\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}^{\zeta}_{0,{\mathbf{f}},\operatorname{\mathbf{d}}}\right)$ is a split surjective map for every $n$ by Lemma \[splitinj\].  Applying ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}$ to this map, we deduce that the top map in the commutative diagram $$\xymatrix{
{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\mathcal{IC}}_{X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\ar[d]\ar[rr]^-{}&&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\operatorname{\mathcal{H}}\left(\hat{p}_{\operatorname{\mathbf{d}},*}^{\zeta}j_*{\mathcal{IC}}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\ar[d]\\
\operatorname{\mathcal{H}}\left(\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{(1,\operatorname{\mathbf{d}})}^{\zeta^{[\mu]}}}}{\mathcal{IC}}_{X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\ar[rr]^-{}
&&\operatorname{\mathcal{H}}\left(\hat{p}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta} j_*{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)
}$$ is split surjective in each cohomological degree.  Factoring the map $\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}=p_{\operatorname{\mathbf{d}}}^{\zeta}\circ \pi $ where $\pi \colon X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}\rightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta}$ is the affine fibration induced by the inclusion of quivers $Q\rightarrow Q_{{\mathbf{f}}}$, we obtain from Propositions \[basicfacts\] and \[cvs\] that the leftmost vertical arrow is an isomorphism.  On the other hand, $\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}} j=\pi_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$ and so is projective, so that the rightmost vertical map is an isomorphism too.  It follows that $\Theta_W$, which is given by the bottom map after identifying $$\operatorname{\mathcal{H}}\left(\hat{p}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{(1,\operatorname{\mathbf{d}})}^{\zeta_{{\mathbf{f}}}^{[\mu]}}}}{\mathcal{IC}}_{X_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\cong\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}{\mathcal{IC}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)$$ via homotopy invariance, is a split surjection in each cohomological degree.  The proof that $\Theta_W$ is $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$-linear is as in [@Fran13 Prop.3.4].  Alternatively, linearity follows from the linearity part of Proposition \[canmaps\] below, and the realisation of $\operatorname{\mathcal{H}}\left(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu}\right)$ as the limit of $\operatorname{\mathcal{H}}\left(\mathcal{F}^{\zeta}_{W,{\mathbf{f}},\mu}\right)$ as ${\mathbf{f}}\mapsto\infty$.

\[cyclescor\] The map $$\begin{aligned}
\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})\rightarrow &\operatorname{H}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta})\\
x\mapsto & \operatorname{H}(\cdot^{\zeta}_{W,{\mathbf{f}},\mu})(x,z_0), \end{aligned}$$ where $z_0=1\in\operatorname{H}\left({\mathcal{M}}_{{\mathbf{f}},0}^{\zeta},{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},0}^{\zeta}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},0}^{\zeta}}({\mathbb{Q}})\right)\cong\mathbb{Q}$, is a surjective map of $\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}$-modules. 

We deduce from Proposition \[PCom\] and Theorem \[cyclesprop\] that the map is a split surjection after passing to the associated graded with respect to the perverse filtration, and the result follows. 

To complete our account of the representation theory of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$, we introduce natural morphisms $\operatorname{\mathcal{H}}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta})\rightarrow \operatorname{\mathcal{H}}(\mathcal{F}_{W,{\mathbf{f}}',\mu}^{\zeta})$ for ${\mathbf{f}}'<{\mathbf{f}}$. Consider the inclusion of quivers $Q_{{\mathbf{f}}'}\subset Q_{{\mathbf{f}}}$. We define a functor from the category of $\mathbb{C}Q_{{\mathbf{f}}}$-modules to the category of $\mathbb{C}Q_{{\mathbf{f}}'}$-modules by precomposing with this inclusion, defining the map $$\Phi\colon X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}/G_{\operatorname{\mathbf{d}}}\rightarrow X(Q_{{\mathbf{f}}'})^{\zeta^{[\mu]}_{{\mathbf{f}}'}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}/G_{\operatorname{\mathbf{d}}}.$$ We define $N_{\operatorname{\mathbf{e}}}/G_{\operatorname{\mathbf{d}}}:=\Phi^{-1}\left(X(Q_{{\mathbf{f}}'})^{(\zeta_{{\mathbf{f}}'}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{e}}),\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}/G_{\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}\right)$ where we use again the Harder–Narasimhan stratification $$Y_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta^{[\mu]}_{{\mathbf{f}}'}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}}:=\coprod_{\operatorname{\mathbf{e}}\leq_t \operatorname{\mathbf{d}}}X(Q_{{\mathbf{f}}'})^{(\zeta_{{\mathbf{f}}'}^{(\mu)},\zeta)\operatorname{-ss}}_{(1,\operatorname{\mathbf{e}}),\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}/G_{\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}}.$$ This induces a stratification $Y^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}=\coprod_{\operatorname{\mathbf{e}}\leq_t \operatorname{\mathbf{d}}}N_{\operatorname{\mathbf{e}}}/G_{\operatorname{\mathbf{d}}}$ with open stratum $j\colon N_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}\rightarrow {\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}$. The map $N_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}\rightarrow Y_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}$ is an affine fibration, with section given by extending a $\mathbb{C}Q_{{\mathbf{f}}'}$-module to a $\mathbb{C}Q_{{\mathbf{f}}}$-module by setting the action of all of the arrows in $\mathbb{C}Q_{{\mathbf{f}}}$ that are not in $\mathbb{C}Q_{{\mathbf{f}}'}$ to be the zero map.  We define $\Xi_{{\mathbf{f}},{\mathbf{f}}'}$ to be the following composition of morphisms.  $$\begin{aligned}
&\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\right)\rightarrow \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}j_*{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}|_{N_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}}}{\mathcal{IC}}_{N_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2}\xrightarrow{\cong}\\&\xrightarrow{\cong}\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\otimes {\mathbb{L}}^{{\mathbf{f}}\cdot \operatorname{\mathbf{d}}/2-({\mathbf{f}}\cdot \operatorname{\mathbf{d}}-{\mathbf{f}}'\cdot \operatorname{\mathbf{d}})/2}\xrightarrow{=}\operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}^{\zeta}{\mathfrak{IC}}_{W,{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}\right).\end{aligned}$$

\[canmaps\] The morphism $\Xi_{{\mathbf{f}},{\mathbf{f}}'}\colon \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}\right)\rightarrow \operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}}',\operatorname{\mathbf{d}},*}^{\zeta}{\mathfrak{IC}}_{W,{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}\right)$ is a morphism of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$-modules, which is a split surjection in each cohomological degree. 

We first show $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$-linearity.  The inclusion $X(Q_{{\mathbf{f}}'})^{\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\rightarrow X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ defined by extending by zero induces a map $$\begin{aligned}
\Omega_0\colon &\operatorname{\mathcal{H}}\left(\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\rightarrow\\ &\operatorname{\mathcal{H}}\left(\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}''),*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta_{{\mathbf{f}}'}^{(\mu)}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}}}{\mathcal{IC}}_{X(Q_{{\mathbf{f}}'})^{\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2-({\mathbf{f}}-{\mathbf{f}}')\cdot \operatorname{\mathbf{d}}''/2}\end{aligned}$$ where $\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}$ and $\tau_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}$ are as in (\[modDiag2\]).  We first check the relation $$\label{Co1}
\Omega_0\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}=\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}(\operatorname{id}\boxtimes\Xi_{{\mathbf{f}},{\mathbf{f}}'}).$$ Consider the diagram $$\xymatrix{
X(Q_{{\mathbf{f}}})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''} \ar[r]^-{r_1}& \left( X(Q)_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times Y_{{\mathbf{f}},\operatorname{\mathbf{d}}''}^{\zeta}\right)/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\\
X(Q_{{\mathbf{f}}'})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\ar[u] \ar[r]^-{r'_1}& \left( X(Q)_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times Y_{{\mathbf{f}}',\operatorname{\mathbf{d}}''}^{\zeta}\right)/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\ar[u]
}$$ where $r'_1$ is defined in the same way as $r_1$, defined as in (\[modDiag\]), and the vertical maps are the inclusions.  This diagram is Cartesian, as are the similarly defined diagrams $$\xymatrix{
X(Q_{{\mathbf{f}}})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\ar[r]^-{r_2}&X(Q_{{\mathbf{f}}})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\\
X(Q_{{\mathbf{f}}'})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}\ar[u]\ar[r]^-{r'_2}&X(Q_{{\mathbf{f}}'})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}.\ar[u]
}$$ and $$\xymatrix{
X(Q_{{\mathbf{f}}})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}&\ar[l]_-{h}X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\\
X(Q_{{\mathbf{f}}'})^{\zeta\operatorname{-ss},\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[u]&\ar[l]_-{h'}X(Q_{{\mathbf{f}}'})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[u]
}$$ with $h,h',r_2,r'_2$ defined as in (\[modDiag\]).  Breaking $\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}$ and $\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}$ into their constituent morphisms, the equation (\[Co1\]) breaks into a number of simpler commutativity relations, which are all obtained via commutativity of pullbacks obtained from the above commutative diagrams.  Next consider the commutative diagram $$\label{paral}
\xymatrix{
X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[rr]^-{\iota_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}&&Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[rr]^{r_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}}&&Y_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}}
\\ X(Q_{{\mathbf{f}}'})^{\zeta_{{\mathbf{f}}'}^{(\mu)}\operatorname{-ss}}_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[u]\ar[rr]^-{\iota_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}}&&Y_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[rr]^{r_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}}\ar[u]^{\iota_1}&&Y_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}/G_{\operatorname{\mathbf{d}}}\ar[u]^{\iota_2}
}$$ where again the vertical arrows are the inclusions.  As in (\[modDiag2\]) we define $$\begin{aligned}
s_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}=&r_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\iota_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\\
s_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}=&r_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\iota_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}.\end{aligned}$$ Then the squares in (\[paral\]) are cartesian squares of projective maps, with the leftmost square a transversal intersection.  We define $$\begin{aligned}
X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}''),N}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}=&X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}'')}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}\times_{G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}U_N\\
Z^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}=&X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}\times_{G_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}U_N\\
X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}),N}=&X(Q_{{\mathbf{f}}})^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}})}\times_{G_{\operatorname{\mathbf{d}}}}U_N\end{aligned}$$ with $U_N=\prod_{i\in Q_0}\operatorname{Hom}^{\mathrm{surj}}(\mathbb{C}^N,\mathbb{C}^{\operatorname{\mathbf{d}}_i})$ as before.  Then we define maps $$\begin{aligned}
\gamma_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N}\colon &\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N}}}s_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N,*}{\mathbb{Q}}_{X(Q_{{\mathbf{f}}})_{\operatorname{\mathbf{d}}',(1,\operatorname{\mathbf{d}}''),N}^{\zeta_{{\mathbf{f}}}^{(\mu)}\operatorname{-ss}}}\right)\rightarrow\\&\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N}}}r_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N,*}{\mathbb{Q}}_{Z^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right)\otimes{\mathbb{L}}^{\sum_{a\in Q_1}\operatorname{\mathbf{d}}'_{s(a)}\operatorname{\mathbf{d}}''_{t(a)}}\end{aligned}$$ and $$\begin{aligned}
\delta_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N}\colon &\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N}}}r_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}''),N,*}{\mathbb{Q}}_{Z^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right)\otimes{\mathbb{L}}^{\sum_{a\in Q_1}\operatorname{\mathbf{d}}'_{s(a)}\operatorname{\mathbf{d}}''_{t(a)}}\rightarrow\\
&\operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N,*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},N}}}{\mathbb{Q}}_{X(Q_{{\mathbf{f}}})^{\zeta^{(\mu)}_{{\mathbf{f}}}\operatorname{-ss}}_{(1,\operatorname{\mathbf{d}}),N}}\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')}.\end{aligned}$$ We define maps $\Omega_1$ and $\Omega_2$ in the same way as $\Omega_0$, by restricting along the inclusions $\iota_{1,N}$ and $\iota_{2,N}$ and passing to the limit in cohomology.  From [@Da13 Cor.2.13] we deduce the following identities $$\begin{aligned}
\Omega_1\gamma_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}=\gamma_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\Omega_0\\
\Omega_2\delta_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}=\delta_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\Omega_1.\end{aligned}$$ Now, from the equations $$\begin{aligned}
\Omega_2=&\Xi_{{\mathbf{f}},{\mathbf{f}}'}\\ \beta_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}=&\delta_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\gamma_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\\ \beta_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}=&\delta_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\gamma_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\end{aligned}$$ we deduce $$\begin{aligned}
\operatorname{\mathcal{H}}(\operatorname{\tilde{\cdot}}_{W,{\mathbf{f}}',\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})\circ(\operatorname{id}\boxtimes \Xi_{{\mathbf{f}},{\mathbf{f}}'})=&\beta_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\circ(\operatorname{id}\boxtimes \Xi_{{\mathbf{f}},{\mathbf{f}}'})
\\=&\beta_{\operatorname{\mathbf{d}}',({\mathbf{f}}',\operatorname{\mathbf{d}}'')}\Omega_0\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}
\\=&\Xi_{{\mathbf{f}},{\mathbf{f}}'}\beta_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}\alpha_{\operatorname{\mathbf{d}}',({\mathbf{f}},\operatorname{\mathbf{d}}'')}
\\=&\Xi_{{\mathbf{f}},{\mathbf{f}}'}\operatorname{\mathcal{H}}(\operatorname{\tilde{\cdot}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}),\end{aligned}$$ finishing the proof of $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$-linearity.  The proof of surjectivity is the same as in Theorem \[cyclesprop\]: one starts by showing that the morphism $$\operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}(\mathbb{Q})\right)\rightarrow \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}j_*{\mathcal{IC}}_{N_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)$$ is a split surjection between pure mixed Hodge modules in each cohomological degree, and then uses that $N_{\operatorname{\mathbf{d}}}/G_{\operatorname{\mathbf{d}}}\rightarrow {\mathcal{M}}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}^{\zeta}$ is an affine fibration to deduce that $$\operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},*}^{\zeta}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}(\mathbb{Q})\right)\rightarrow \operatorname{\mathcal{H}}\left(\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}j_*{\mathcal{IC}}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}}',\operatorname{\mathbf{d}}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{-({\mathbf{f}}-{\mathbf{f}}')\cdot \operatorname{\mathbf{d}}/2}$$ is a split surjection in each cohomological degree, and then proceeds as in the proof of Theorem \[cyclesprop\]. 

We define the morphism of $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})$-modules $\operatorname{H}(\Xi_{{\mathbf{f}},{\mathbf{f}}'})\colon \operatorname{H}(\mathcal{F}_{W,{\mathbf{f}},\mu}^{\zeta})\rightarrow \operatorname{H}(\mathcal{F}_{W,{\mathbf{f}}',\mu}^{\zeta})$ in the same way as $\Xi_{{\mathbf{f}},{\mathbf{f}}'}$. Then after passing to the associated graded with respect to the perverse filtration, $\operatorname{H}(\Xi_{{\mathbf{f}},{\mathbf{f}}'})$ is a split surjection, as it is equal to $\operatorname{\mathcal{H}}\left(\dim_*\Xi_{{\mathbf{f}},{\mathbf{f}}'}\right)$. It follows that $\operatorname{H}(\Xi_{{\mathbf{f}},{\mathbf{f}}'})$ is surjective.  Putting together Lemma \[appLem\], Theorem \[cyclesprop\], Corollary \[cyclescor\] and Proposition \[canmaps\], we deduce Theorem \[repthm\]. 

The perverse associated graded Hopf algebra {#PBWsec}
===========================================

Proof of Theorem \[qea\]
------------------------

Let $\zeta\in \mathbb{H}_+^{Q_0}$ be a stability condition, and let $\mu\in(\infty,\infty)$ be a slope - we make no genericity assumption on $\zeta$ for now.  Consider again the algebra from the end of Section \[relc\] $$\operatorname{H}\left(\mathcal{A}_{W,\mu}^{\zeta}\right)=\operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\mu})_*{\mathfrak{IC}}_{W,\mu}^{\zeta}\right).$$ By [@Da13 Thm.5.11] this algebra carries a localised bialgebra structure, in the sense that for all decompositions $\operatorname{\mathbf{d}}=\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''$, with $\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}$, there are maps $$\begin{aligned}
\label{coprodint}
\operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}})_*{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta}\right)\rightarrow \left(\operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}'})_*{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\right)\boxtimes_+^{\operatorname{tw}} \operatorname{\mathcal{H}}\left((\dim\circ p^{\zeta}_{\operatorname{\mathbf{d}}''})_*{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}\right)\right)[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\end{aligned}$$ where $$\begin{aligned}
\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}:=&\prod_{i,j\in Q_0}\prod_{1\leq t'\leq \operatorname{\mathbf{d}}'(i)}\prod_{1\leq t''\leq \operatorname{\mathbf{d}}''(j)}(y_{j,t''}-x_{i,t'})\\
\in &\operatorname{H}_{G_{\operatorname{\mathbf{d}}'}}(\operatorname{pt})\otimes \operatorname{H}_{G_{\operatorname{\mathbf{d}}''}}(\operatorname{pt})\\
=&\mathbb{C}[x_{1,1},\ldots,x_{1,\operatorname{\mathbf{d}}'_1},\ldots,x_{n,1},\ldots,x_{n,\operatorname{\mathbf{d}}'_n}]^{\operatorname{Sym}_{\operatorname{\mathbf{d}}'}}\otimes\\&\otimes \mathbb{C}[y_{1,1},\ldots,y_{1,\operatorname{\mathbf{d}}''_1},\ldots,y_{n,1},\ldots,y_{n,\operatorname{\mathbf{d}}''_n}]^{\operatorname{Sym}_{\operatorname{\mathbf{d}}''}}\end{aligned}$$ with $Q_0=\{1,\ldots,n\}$. These maps are required to satisfy the natural compatibility condition with the multiplication, see [@Da13 Def.5.3].  We will describe this comultiplication in a little more detail, for further details see [@Da13].  Consider the proper map $s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\colon {\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\rightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$. We form the map $$\gamma\colon \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta})\rightarrow \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''},{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q}))\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}$$ given as the tensor product with ${\mathbb{L}}^{(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}$ of the limit as $N\mapsto\infty$ of the composition of maps $$\begin{aligned}
\label{sagain}
&\operatorname{\mathcal{H}}\left(\dim_*p^{\zeta}_{\mu,N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}_{\operatorname{\mathbf{d}},N}^{\zeta}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}},N}^{\zeta\operatorname{-ss}}}\right)\rightarrow \operatorname{\mathcal{H}}\left(\dim_*p^{\zeta}_{\mu,N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}},N}}}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\mathbb{Q}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right)\rightarrow\\&\rightarrow \operatorname{\mathcal{H}}\left(\dim_*p^{\zeta}_{\mu,N,*}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}}{\mathbb{Q}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right),\nonumber\end{aligned}$$ where $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}},N}$ and $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}$ are as in (\[XNdef\]) and (\[XXNdef\]) respectively.  Considering instead the composition of maps $$\begin{aligned}
\label{alterng}
&\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}_{\operatorname{\mathbf{d}},N}^{\zeta}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}},N}^{\zeta\operatorname{-ss}}}\right)\right)\rightarrow \operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}},N}}}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\mathbb{Q}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right)\right)\rightarrow\\&\rightarrow \operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,N,*}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}}{\mathbb{Q}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}}\right)\right)\nonumber\end{aligned}$$ we deduce that $\gamma$ respects the perverse filtration, i.e. $$\gamma\left(\operatorname{\mathfrak{P}}_s\left(\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta})\right)\right)\subset \operatorname{\mathfrak{P}}_{s}\left(\operatorname{H}\left({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''},{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\right).$$ Composing with the isomorphism $$\begin{aligned}
\Gamma\colon &\operatorname{H}\left({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''},{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}(\mathbb{Q})\right)\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\rightarrow \\&\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\otimes\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2+(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\end{aligned}$$ we likewise deduce $$\begin{aligned}
\nonumber
&\Gamma\circ\gamma\left(\operatorname{\mathfrak{P}}_s\left(\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta})\right)\right)\subset
\\\nonumber& \operatorname{\mathfrak{P}}_s\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\otimes\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2+(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}')/2}\right)\\\nonumber
& =\operatorname{\mathfrak{P}}_s\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}}\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}))\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')}\right)\\\label{downshift}
& =\operatorname{\mathfrak{P}}_{s-2(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')}\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}}\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}))\right),\end{aligned}$$ since $\Gamma$ lifts to a morphism of monodromic mixed Hodge modules on ${\mathcal{M}}^{\zeta\operatorname{-ss}}$. The final equality (\[downshift\]) does not respect the cohomological degree, or the mixed Hodge structure.  We define $$\mathfrak{E}_{1,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}:=\prod_{a\in Q_1}\prod_{1\leq l'\leq \operatorname{\mathbf{d}}'_{s(a)}}\prod_{1\leq l''\leq \operatorname{\mathbf{d}}''_{t(a)}}(x_{s(a),l'}-y_{t(a),l''})$$ and $$\mathfrak{E}_{0,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}:=\prod_{i\in Q_0}\prod_{1\leq l'\leq \operatorname{\mathbf{d}}'_i}\prod_{1\leq l''\leq \operatorname{\mathbf{d}}''_i}(x_{i,l'}-y_{i,l''}).$$

Multiplication by $\mathfrak{E}^{-1}_{1,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\mathfrak{E}_{0,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ defines a map $$\begin{aligned}
\beta^*\colon &\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}}\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\rightarrow \\&  \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}}\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})[\mathfrak{L}^{-1}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}].\end{aligned}$$ The comultiplication is defined [@Da13 Cor.5.9] by $$\begin{aligned}
 \Delta_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}\colon &\mathcal{A}_{Q,W,\operatorname{\mathbf{d}}}^{\zeta}\rightarrow \left(\mathcal{A}_{Q,W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes^{\operatorname{tw}}_+ \mathcal{A}_{Q,W,\operatorname{\mathbf{d}}''}^{\zeta}\right)[\mathfrak{L}^{-1}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}]
\\ 
\Delta_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta}:=&(\cdot\mathfrak{E}^{-1}_{1,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\mathfrak{E}_{0,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})\circ \Gamma\circ\gamma.   \end{aligned}$$ The comultiplication respects cohomological degree, and lifts to a morphism in the category of monodromic mixed Hodge structures.  For a dimension vector $\operatorname{\mathbf{e}}$, we define $\lvert \operatorname{\mathbf{e}}\rvert=\sum_{i\in Q_0}\operatorname{\mathbf{e}}_i$. So $\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ is of cohomological degree $2\lvert\operatorname{\mathbf{d}}'\rvert \lvert \operatorname{\mathbf{d}}''\rvert$. We extend the perverse filtration to the right hand side of (\[coprodint\]) by setting $$\begin{aligned}
&\operatorname{\mathfrak{P}}_{s}\left(\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right)=\\&\sum_{n\geq 0}\operatorname{\mathfrak{P}}_{s+2\lvert\operatorname{\mathbf{d}}''\rvert\lvert\operatorname{\mathbf{d}}'\rvert n}\left(\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-n}.\end{aligned}$$ By [@Da13 Prop.4.1], the natural map $$\begin{aligned}
&\left(\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)\xrightarrow{f_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\\& \left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\end{aligned}$$ is an inclusion, since the operation $\cdot\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ is injective on the domain.  By definition, $f_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ preserves the perverse filtration. 

\[HCM2\] The group $\operatorname{\mathfrak{P}}_s\left(\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right)$ consists of those elements $a\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-n}$ such that $a\in \operatorname{\mathfrak{P}}_{s+2n\lvert\operatorname{\mathbf{d}}'\rvert\lvert\operatorname{\mathbf{d}}''\rvert}\left(\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)\right)$.

One inclusion is clear.  On the other hand, say we can write $$a=\sum_{i=0}^{n}a_i\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-i}$$ with $a_i\in \operatorname{\mathfrak{P}}_{s+2i\lvert\operatorname{\mathbf{d}}'\rvert\lvert\operatorname{\mathbf{d}}''\rvert}\left(\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)\right)$. Then by Lemma \[HCM\], each term $a_i\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{n-i}\in \operatorname{\mathfrak{P}}_{s+2n\lvert\operatorname{\mathbf{d}}'\rvert\lvert\operatorname{\mathbf{d}}''\rvert}\left(\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)\right)$ and the lemma follows. 

By Lemma \[HCM\] and Proposition \[HCM2\], multiplication by $\mathfrak{E}_{1,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}\cdot\mathfrak{E}_{0,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ induces a map $$\begin{aligned}
\nonumber&\operatorname{\mathfrak{P}}_s\left(\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right)\rightarrow \\ \label{upshift}
&\operatorname{\mathfrak{P}}_{s+2(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')}\left(\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right).\end{aligned}$$ Combining the shifts in perverse degree in (\[upshift\]) and (\[downshift\]), we deduce that $$\Delta^{\zeta}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\left(\operatorname{\mathfrak{P}}_s (\operatorname{H}({\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}}^{\zeta}))\right)\subset \operatorname{\mathfrak{P}}_s\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}}\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right).$$ The proof of the following is basically the original proof of the Atiyah–Bott lemma of [@AtBo83], adapted to deal with vanishing cycles and passage to the associated graded of the perverse filtration. 

\[PAB\] The map $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(f_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})$ is injective. 

We abbreviate $\operatorname{\mathfrak{P}}_i=\operatorname{\mathfrak{P}}_i\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)$. It is sufficient to show that the degree $2\lvert\operatorname{\mathbf{d}}'\rvert \lvert\operatorname{\mathbf{d}}''\rvert$ map $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})$ is injective.  For then if $a$ in the domain of $f_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ lies in $\operatorname{\mathfrak{P}}_{i+1}\setminus \operatorname{\mathfrak{P}}_i$, then for all $r\in\mathbb{N}$, $(\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})^r\cdot a\notin \operatorname{\mathfrak{P}}_{i+2r|\operatorname{\mathbf{d}}'||\operatorname{\mathbf{d}}''|}$, so that $$f_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}(a)\notin \left(\sum_{r\geq 0}\operatorname{\mathfrak{P}}_{i+2\lvert\operatorname{\mathbf{d}}'\rvert \lvert\operatorname{\mathbf{d}}''\rvert r}\left(\operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta})\boxtimes_+^{\operatorname{tw}} \operatorname{H}({\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-r}\right)$$ and $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(f_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})(a)\neq 0$.

Consider the subgroup ${\mathbb{G}}_m\cong T\subset G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}$ given by the embedding $$z\mapsto \left( (z^{\lvert\operatorname{\mathbf{d}}''\rvert}\operatorname{id}_{\mathbb{C}^{\operatorname{\mathbf{d}}'_i}})_{i\in Q_0},(z^{-\lvert\operatorname{\mathbf{d}}'\rvert}\operatorname{id}_{\mathbb{C}^{\operatorname{\mathbf{d}}''_i}})_{i\in Q_0}\right)$$ and let $P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}:=(G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''})/T$. Note that $T$ acts trivially on $X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}$ and furthermore $T$ acts trivially on the linearization (\[charDef\]), so that the linearization of the $G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}$-action lifts to a linearization of the $P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ action.  Let $$p'_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\colon (X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''})/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\rightarrow {\mathcal{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times {\mathcal{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}$$ be the map to the GIT quotient.  Via the Cartesian square $$\xymatrix{
{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}\times {\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}\ar[r]\ar[d]&\operatorname{pt}/(G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''})\ar[d]\\
(X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''})/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\ar[r]&\operatorname{pt}/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}
}$$ we obtain the isomorphism $$\begin{aligned}
&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}\right)\cong\\& \operatorname{\mathcal{H}}\left(p'_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes_{\operatorname{H}_{P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\operatorname{pt})}\operatorname{H}_{G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}}(\operatorname{pt})\otimes {\mathbb{L}}^{1/2}.\end{aligned}$$ where $f$ is the function induced by ${\mathrm{Tr}(W)}_{\operatorname{\mathbf{d}}'}\boxplus {\mathrm{Tr}(W)}_{\operatorname{\mathbf{d}}''}$ on $(X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''})/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$. After picking an isomorphism $\lambda\colon \operatorname{H}_{G_{\operatorname{\mathbf{d}}'}\times G_{\operatorname{\mathbf{d}}''}}(\operatorname{pt})\cong \operatorname{H}_{P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\operatorname{pt})\otimes \operatorname{H}_{T}(\operatorname{pt})$ we obtain the isomorphism $$\begin{aligned}
&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}\right)\cong \operatorname{\mathcal{H}}\left(p'_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes \operatorname{H}({\mathcal{IC}}_{\operatorname{pt}/T}(\mathbb{Q})).\end{aligned}$$ Let $$\begin{aligned}
D^i=&\operatorname{\mathcal{H}}^{\geq i}\left(p'_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{f}}{\mathcal{IC}}_{X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times X^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}/P_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}(\mathbb{Q})\right)\otimes \operatorname{H}({\mathcal{IC}}_{\operatorname{pt}/T}(\mathbb{Q}))\\&\subset \operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\end{aligned}$$ and let $E^i_j=D^i\cap \operatorname{\mathfrak{P}}_j$. We remark, though we do not use the fact, that the filtration $D$, and hence the filtration $E$, does not depend on $\lambda$. Consider the associated bigraded object $\operatorname{Gr}_E\left(\operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\right)$. Multiplication by $\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}$ preserves $D$ degree, and so induces a map $$\operatorname{Gr}_{E}(\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})\colon \operatorname{Gr}_{E,j}^i\operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta})\rightarrow \operatorname{Gr}_{E,j+|\operatorname{\mathbf{d}}'||\operatorname{\mathbf{d}}''|}^i\operatorname{\mathcal{H}}(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}'}^{\zeta}\boxtimes p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}''}^{\zeta}).$$ Specifically, the map is given by multiplication by the $T$-equivariant Euler characteristic of $X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\rightarrow X_{\operatorname{\mathbf{d}}'}\times X_{\operatorname{\mathbf{d}}''}$, which is nonzero since the vector bundle has no sub bundle with trivial $T$-weight.  It follows that $\operatorname{Gr}_{E}(\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})$ is injective, so that $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\cdot \mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})$ is injective too. 

The triple $\left(\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})),\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\tilde{\ast}}^{\zeta}_{W,\mu})),\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\Delta_{W,\mu}^{\zeta})\right)$ defines a localised bialgebra in the sense of [@Da13 Def.5.3]. 

Now assume that $\zeta$ is $\mu$-generic.  By Corollary \[inccor\] and Proposition \[PCoha\], there is a natural inclusion in ${\mathcal{D}^{\geq}}(\operatorname{MMHM}(\Lambda^{\zeta}_{\mu}))$ $$\bigoplus_{\operatorname{\mathbf{d}}\in \Lambda_{\mu}^{\zeta}}\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\hookrightarrow\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu}))$$ where $$\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}:=\operatorname{H}({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}},{\mathcal{DT}}_{W,\operatorname{\mathbf{d}}}^{\zeta}).$$

\[primitivity\] If $\zeta$ is a $\mu$-generic stability condition, then for $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$ the subspace $\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$ is a primitive subspace in $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}))$, i.e. the composition of maps $$\begin{aligned}
&\bigoplus_{\operatorname{\mathbf{d}}\in \Lambda_{\mu}^{\zeta}}\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\hookrightarrow\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu}))\xrightarrow{\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\Delta^{\zeta}_{W,\mu})}\\&\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}}\operatorname{Gr}_{\operatorname{\mathfrak{P}}}\left((\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\operatorname{\mathbf{d}}'})\boxtimes^{\operatorname{tw}}_+\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\operatorname{\mathbf{d}}''}))[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right)
\twoheadrightarrow \bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}\setminus\{0\}}\operatorname{Gr}_{\operatorname{\mathfrak{P}}}\left((\operatorname{\mathcal{A}}^{\zeta}_{W,\operatorname{\mathbf{d}}'}\boxtimes^{\operatorname{tw}}_+\operatorname{\mathcal{A}}^{\zeta}_{W,\operatorname{\mathbf{d}}''})[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\right)\end{aligned}$$ is zero. 

We need to show that for decompositions $\operatorname{\mathbf{d}}=\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''$, with $\operatorname{\mathbf{d}}'\neq 0\neq \operatorname{\mathbf{d}}''$, the image of $\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})$ under the map $$\begin{aligned}
\label{int4}
&\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\Delta^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})\colon \operatorname{H}\left({\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}},\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}\right)\right)\rightarrow \\ \nonumber&\left(\operatorname{H}\left({\mathcal{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}},\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}'}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}'}^{\zeta\operatorname{-ss}}}\right)\right)\boxtimes_{+}^{\operatorname{tw}} \operatorname{H}\left({\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''},\operatorname{\mathcal{H}}\left(p^{\zeta}_{\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}''}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}\right)\right)\right)[\mathfrak{L}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{-1}]\end{aligned}$$ is zero.  We assume that ${\mathcal{M}}^{\zeta\operatorname{-st}}_{\operatorname{\mathbf{d}}}\neq \emptyset$, or the statement is trivial, for then by definition $\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}=0$. In terms of the commutative diagram $$\label{int5}
\xymatrix{
\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}\right)\right)\ar[r]^-{\gamma''} &\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}}{\mathbb{Q}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\right)\right)\\
\operatorname{\mathcal{H}}\left(\dim_*{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}\right)\right)\ar[r]^-{\gamma'} \ar[u]^{{\mathbb{L}}^{-(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\otimes\operatorname{\mathcal{H}}(\dim_*\nu_{\operatorname{\mathbf{d}}})}&\operatorname{\mathcal{H}}\left(\dim_*{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\mathbb{Q}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\right)\right)\ar[u]^{{\mathbb{L}}^{-(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\otimes\operatorname{\mathcal{H}}(\dim_*\nu_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''})}\\
\operatorname{\mathcal{H}}\left(\dim_* {\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}({\mathbb{Q}})\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\ar[u]^{\iota}
}$$ where $\gamma''=\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\gamma)\otimes{\mathbb{L}}^{-(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}$, it is sufficient to show that $$\gamma''\circ({\mathbb{L}}^{-(\operatorname{\mathbf{d}},\operatorname{\mathbf{d}})/2}\otimes\operatorname{\mathcal{H}}(\dim_*\nu_{\operatorname{\mathbf{d}}}))\circ\iota=0.$$ On the other hand, the map $\gamma'\iota$ is obtained by applying $\operatorname{\mathcal{H}}\left(\dim_*{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}^{\zeta}}}\right)$ to a map $$\begin{aligned}
\label{nearmiss}
{\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}}({\mathbb{Q}})\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\rightarrow \operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}s_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',*}{\mathbb{Q}}_{{\mathfrak{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}}\right)\end{aligned}$$ which is necessarily zero, since for every cohomological degree, the domain and the target of (\[nearmiss\]) are pure complexes of mixed Hodge modules with distinct strict support. 

Let $\mathcal{P}^{\zeta}_{W,\mu}\subset \operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu}))$ be the subalgebra generated by $\bigoplus_{\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}}\operatorname{DT}_{W,\operatorname{\mathbf{d}}}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$. Since $\mathcal{P}^{\zeta}_{W,\mu}$ is generated by primitive elements of the localised Hopf algebra, it follows from Lemma \[PAB\] that the localised bialgebra structure on $\mathcal{P}^{\zeta}_{W,\mu}$ lifts to an honest bialgebra structure.  We denote by $\overline{\Delta}_{W,\mu}^{\zeta}\colon \mathcal{P}^{\zeta}_{W,\mu}\rightarrow \mathcal{P}^{\zeta}_{W,\mu}\boxtimes_+ \mathcal{P}^{\zeta}_{W,\mu}$ the induced coproduct. 

\[trueHopf\] Assume that $\zeta$ is $\mu$-generic.  The quadruple $$\left(\mathcal{P}_{W,\mu}^{\zeta},\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})\right), \overline{\Delta}^{\zeta}_{W,\mu},\mathcal{P}_{W,0}\right).$$ extends to a Hopf algebra. 

All that is missing is a compatible antipode, but existence and uniqueness of an antipode is a formal consequence of connectedness of the algebra. 

Consider the inclusion $$\iota \colon \operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\hookrightarrow \operatorname{Free}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}).$$ Retaining the assumption that $\zeta$ is $\mu$-generic, the relative cohomological Hall algebra map $\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})\colon \operatorname{\mathcal{H}}(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta})\boxtimes_{\oplus}\operatorname{\mathcal{H}}(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta})\rightarrow \operatorname{\mathcal{H}}(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta})$ along with the inclusions $$\operatorname{Free}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\rightarrow \operatorname{Free}_{\boxtimes_{\oplus}}\left(\operatorname{\mathcal{H}}(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta})\right)$$ and $\iota$ induce a map $$\label{GaDef}
\Gamma_{W,\mu}^{\zeta}\colon \operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\rightarrow \operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})$$

\[inj\] The map $\Gamma^{\zeta}_{W,\mu}$ is an injection. 

We start by assuming $W=0$. We first claim that the map $$\label{pron}
\Psi\colon \operatorname{Sym}_{\boxtimes_{+}}(\operatorname{H}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}},{\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}))\rightarrow \mathcal{P}^{\zeta}_{0,\mu}$$ is an injection.  For this, consider the non-counital coproduct $\overline{\Delta}^{\zeta}_{0,\mu,\operatorname{red}}$ induced by $\overline{\Delta}^{\zeta}_{0,\mu}$ on the nonunital algebra $(K=\ker(\mathcal{P}^{\zeta}_{0,\mu}\rightarrow \mathcal{P}^{\zeta}_{0,0}),\operatorname{H}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})\Large{|_K})$. Let $\alpha$ be an element of the left hand side of (\[pron\]), and let $n$ be the maximum number such that the $n$th component of $\alpha$ in the decomposition $$\operatorname{Sym}_{\boxtimes_{+}}\left(\operatorname{H}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}},{\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\right):=\bigoplus_{n}\operatorname{Sym}^n_{\boxtimes_{+}}\left(\operatorname{H}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}},{\mathcal{DT}}_{0,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\right)$$ is nonzero.  We may assume that $n> 0$, otherwise the image of $\alpha$ under (\[pron\]) is a scalar multiple of the unit of $\operatorname{H}(\operatorname{\mathcal{A}}_{0,\mu})$. Then $\overline{\Delta}_{0,\mu,\operatorname{red}}^{\zeta,n}(\Psi(\alpha))=\iota(\alpha)\neq 0$, proving the claim.  On the other hand, $\operatorname{H}(\Gamma_{0,\mu}^{\zeta})$ factors as $i\Psi$, where $i\colon\mathcal{P}^{\zeta}_{0,\mu}\hookrightarrow\operatorname{\mathcal{H}}(\mathcal{A}_{0,\mu}^{\zeta})$ is the inclusion, and so $\operatorname{H}(\Gamma_{0,\mu}^{\zeta})$ is an injection.  Still assuming $W=0$, in each degree, the left and right hand sides of (\[GaDef\]) are direct sums of shifts of simple perverse sheaves ${\mathcal{IC}}_{Z}(\mathbb{Q})$ for $Z\in{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$ locally closed subvarieties, and $\mathbb{Q}$ the trivial local system on $Z$. More precisely, the $Z$ all have the form $\operatorname{Sym}^{\circ,a_1}X_{\operatorname{\mathbf{d}}_1}\times \ldots \operatorname{Sym}^{\circ,a_i}X_{\operatorname{\mathbf{d}}_i}$ for decompositions $\operatorname{\mathbf{d}}=\sum_{n=1}^i a_n\operatorname{\mathbf{d}}_n$, and $\operatorname{Sym}^{\circ,a_n} X_{\operatorname{\mathbf{d}}_n}$ defined to be $\operatorname{Sym}^{a_n} X_{\operatorname{\mathbf{d}}_n}$ minus the big diagonal, and we have $$\operatorname{H}({\mathcal{IC}}_Z(\mathbb{Q}))=\bigotimes_{n=1}^i\operatorname{Sym}^{a_i}\left(\operatorname{IC}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}_i}}(\mathbb{Q})\right).$$ Each $\operatorname{H}({\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}_i}}(\mathbb{Q}))$ is nonzero: for instance in degree zero, the underlying vector space has a basis indexed by path components of ${\mathcal{M}}^{\zeta\operatorname{-st}}_{\operatorname{\mathbf{d}}_i}$, i.e. $\operatorname{H}^0({\mathcal{IC}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}_i}})\cong\mathbb{Q}$, from which it follows that each $\operatorname{H}({\mathcal{IC}}_{Z}(\mathbb{Q}))\neq 0$. It follows from the injectivity of (\[pron\]) that $\Gamma_{0,\mu}^{\zeta}$ is an injection, since it is a morphism between direct sums of simple objects in a semisimple category, and it is an injection after passing to hypercohomology.  Since it is a map of pure mixed Hodge modules, it is in addition split injective, and so we deduce that $\Gamma^{\zeta}_{0,\mu}$ remains injective after applying ${\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}}}$. Now the result follows from Lemma \[cohacom\], which gives the commutativity of $$\xymatrix{
{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}}}\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}^{\zeta}_{0,\mu}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right) \ar[d]^{\nu}\ar[rr]^-{{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\Gamma_{0,\mu}}&&{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\mu}^{\zeta}}}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{0,\mu}\right)\ar[d]^{\nu}\\
\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}^{\zeta}_{W,\mu}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\ar[rr]^-{\Gamma_{W,\mu}}&&\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}^{\zeta}_{W,\mu}\right). 
}$$

\[zPBW\] The natural map obtained from the relative cohomological multiplication and the natural inclusions of Corollary \[inccor\] $$\Gamma_{W,\mu}\colon \operatorname{Sym}_{\boxtimes_{\oplus}}({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\rightarrow \operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)$$ is an isomorphism in ${\mathcal{D}^{\geq}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))$.

From Lemma \[inj\] we know already that the map is an inclusion, in particular, in the case $W=0$. But by Theorem \[weakPBW\], if $W=0$, the left and the right hand side are also pure mixed Hodge modules with the same class in the Grothendieck group, so that $$\Gamma_{0,\mu}\colon \operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}^{\zeta}_{0,\mu}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\rightarrow \operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{0,\mu}^{\zeta}\right)$$ is an isomorphism.  Again applying Lemma \[cohacom\] we are done. 

Combining Theorem \[zPBW\], Corollary \[trueHopf\] and Proposition \[primitivity\], we deduce that $$\left(\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}\left( p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)\right),\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})\right), \overline{\Delta}^{\zeta}_{W,\mu},\operatorname{\mathcal{H}}\left(\dim_*\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})\right)_0\right)$$ is a connected unital Hopf algebra that is generated by the primitive sub monodromic mixed Hodge module ${\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$, and is therefore the universal enveloping algebra of its Lie algebra of primitive elements.  This completes the proof of Theorem \[qea\]. 

Proof of Theorem \[strongPBW\] {#CWCF}
------------------------------

We finish this section by proving Theorem \[strongPBW\], the cohomological wall crossing theorem relating different PBW bases for the cohomological Hall algebra $\operatorname{\mathcal{H}}(p_*{\mathfrak{IC}}_W)$. It turns out that almost all of the hard work goes into proving that $\operatorname{\mathcal{H}}(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta})$ admits a PBW basis, i.e. Theorem \[zPBW\]. 

Fix a generic stability condition $\zeta$. By Corollary \[inccor\] there are canonical split inclusions ${\mathcal{DT}}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\rightarrow \operatorname{\mathcal{H}}\left(p_{\mu,*}^{\zeta}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)$ giving rise to split inclusions $q^{\zeta}_{\mu,*}{\mathcal{DT}}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\rightarrow q_{\mu,*}^{\zeta}\operatorname{\mathcal{H}}\left(p^{\zeta}_{\mu,*}{\mathfrak{IC}}_{W,\mu}^{\zeta}\right)$. By the proof of Theorem \[gcDT\], the distinguished triangles (\[HNtri\]) split, and picking a splitting for each of the diagrams gives rise to a specific isomorphism (\[gwPBWnc\]), and in particular, for each slope $\mu$, an embedding of $q^{\zeta}_{\mu,*}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$ inside $\operatorname{\mathcal{H}}\left(p_*{\mathfrak{IC}}_W\right)$. We fix this embedding, in writing the main theorem, where for simplicity we also fix an isomorphism $\operatorname{\mathcal{H}}(q^{\zeta}_{\mu,*}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}})\cong q^{\zeta}_{\mu,*}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}$, which we are free to do via the decomposition theorem, stated as in Proposition \[basicfacts\](\[vanDec\]). 

\[sPBW\] If $\zeta$ is a generic stability condition, the map $$\boxtimes_{\oplus,\infty \xrightarrow{\mu}-\infty}^{\operatorname{tw}} \left(\operatorname{Sym}_{\boxtimes_{\oplus}}\left(q_{\mu,*}^{\zeta}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\xrightarrow{\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_W)} \operatorname{\mathcal{H}}\left(p_*{\mathfrak{IC}}_W\right)$$ is an isomorphism. 

Fix a dimension vector $\operatorname{\mathbf{d}}$, and let $\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$ be the set of Harder–Narasimhan types summing to $\operatorname{\mathbf{d}}$, endowed with a total order order refining Reineke’s order as in the proof of Theorem \[gcDT\].  Then since each of the distinguished triangles (\[HNtri\]) are split, there is a canonical filtration $G$, indexed by $\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}$, on $$\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},*}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}_{\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}}({\mathbb{Q}})\right)\cong{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}}}\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},*}{\mathcal{IC}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}}({\mathbb{Q}})\right)$$ defined by setting $$G_{\overline{\operatorname{\mathbf{d}}}}:={\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}}}}\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},*}i_{\leq \overline{\operatorname{\mathbf{d}}}}{\mathcal{IC}}_{{\mathfrak{M}}_{\leq \overline{\operatorname{\mathbf{d}}}}}({\mathbb{Q}})\right).$$ After giving the domain of the map $$\Lambda\colon\bigoplus_{(\operatorname{\mathbf{d}}^1,\ldots,\operatorname{\mathbf{d}}^g)\in\operatorname{HN}^{\geq}_{\operatorname{\mathbf{d}}}}\left(q^{\zeta}_{\operatorname{\mathbf{d}}_1,*}\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}}^1,*}^{\zeta}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}_1}^{\zeta}\right)\boxtimes^{\operatorname{tw}}_{\oplus}\ldots\boxtimes^{\operatorname{tw}}_{\oplus}q^{\zeta}_{\operatorname{\mathbf{d}}_g,*}\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}}^g,*}^{\zeta}{\mathfrak{IC}}_{W,\operatorname{\mathbf{d}}_g}^{\zeta}\right)\right)\xrightarrow{\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_W)} \operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_W\right)$$ the filtration induced by the order on $\operatorname{HN}_{\operatorname{\mathbf{d}}}^{\geq}$, the associated graded with respect to the filtration is the identity map, and the result follows by Theorem \[zPBW\]. 

For a generic stability condition $\zeta$ there exist embeddings $\operatorname{DT}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\subset \operatorname{H}(\operatorname{\mathcal{A}}_{W})$ such that the induced map $$\boxtimes_{\oplus,\infty \xrightarrow{\mu}-\infty}^{\operatorname{tw}}\left(\operatorname{Sym}_{\boxtimes_+}\left(\operatorname{DT}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\xrightarrow{\operatorname{H}(\operatorname{\tilde{\ast}}_W)}\operatorname{H}(\operatorname{\mathcal{A}}_W)$$ is an isomorphism. 

By Theorem \[sPBW\] and Proposition \[PCoha\], the corollary is true after replacing $\operatorname{H}(\operatorname{\mathcal{A}}_W)$ with the perverse associated graded $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W}))=\operatorname{\mathcal{H}}(\dim_*\operatorname{\mathcal{H}}(p_*{\mathfrak{IC}}_W))$. Then for any lift of the embedding $\operatorname{DT}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\subset \operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\operatorname{\mathcal{A}}_{W}))$ to an embedding $\operatorname{DT}^{\zeta}_{W,\mu}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\subset \operatorname{H}(\operatorname{\mathcal{A}}_{W})$, the result follows. 

Special CoHAs
-------------

Let $(Q,W)$ be a quiver with potential, let $\mu\in (-\infty,\infty)$ be a $\zeta$-generic stability condition for some stability condition $\zeta$. Without modifying the above constructions, we do not expect to get a nontrivial Lie bracket on the space of primitive elements $\operatorname{DT}_{W,\mu}^{\zeta}\otimes \operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})$ on the associated perverse associated Hopf algebra $\operatorname{Gr}_{\operatorname{\mathfrak{P}}}(\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta}))$; essentially the same argument as Proposition \[primitivity\] demonstrates vanishing of the Lie bracket.  The goal of this section is firstly to prove the categorified integrality theorem and wall crossing formula for DT invariants of ‘special’ $\mathbb{C}Q$-modules, and secondly to provide one framework for producing nontrivial Lie brackets. 

We assume that for every $\operatorname{\mathbf{d}}\in\mathbb{N}^{Q_0}$, we have locally closed subvarieties $$l^{\zeta}_{\operatorname{\mathbf{d}}}\colon {\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\hookrightarrow {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}$$ forming a submonoid.  We denote by $\tilde{l}_{\operatorname{\mathbf{d}}}^{\zeta}\colon {\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\operatorname{Sp},\zeta\operatorname{-ss}}\hookrightarrow{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}$ the associated inclusion of stacks, where ${\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\operatorname{Sp},\zeta\operatorname{-ss}}=(p^{\zeta}_{\operatorname{\mathbf{d}}})^{-1}({\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}})$. We assume that the diagram $$\xymatrix{
{\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times{\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar@{^(->}[d]_{l^{\zeta}_{\operatorname{\mathbf{d}}'}\times l^{\zeta}_{\operatorname{\mathbf{d}}''}}\ar[r]^-{\oplus}&{\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\ar@{^(->}[d]^{l^{\zeta}_{\operatorname{\mathbf{d}}}}\\
{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}'}\times {\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}''}\ar[r]^-{\oplus}&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}
}$$ is Cartesian. 

\[SpLemma\] Let $\operatorname{\mathbf{d}}\in\Lambda_{\mu}^{\zeta}$. Let ${\mathbf{f}}\in\mathbb{N}^{Q_0}$ be a framing vector, and let $$\xymatrix{
{\mathcal{M}}^{\operatorname{Sp},\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\ar[r]^{l^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\ar[d]_{\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}|_{{\mathcal{M}}^{\operatorname{Sp},\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}&{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\ar[d]^{\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\\
{\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}\ar[r]^{l^{\zeta}_{\operatorname{\mathbf{d}}}}&{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\operatorname{\mathbf{d}}}
}$$ be the obvious Cartesian diagram.  Then there are isomorphisms $$\label{ft1}
\operatorname{\mathcal{H}}^n\left(l_{\operatorname{\mathbf{d}},*}^{\zeta}l_{\operatorname{\mathbf{d}}}^{\zeta,*}\operatorname{\mathcal{H}}\left(p_{\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}\right)\right)\cong \lim_{{\mathbf{f}}\mapsto\infty}\left(\operatorname{\mathcal{H}}^n\left(\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}\tilde{l}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\tilde{l}^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}\right)\right).$$ and $$\label{ft2}
\operatorname{H}_c^n({\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}},l_{\operatorname{\mathbf{d}}}^{\zeta,*}\operatorname{\mathcal{H}}(p_{\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathfrak{Tr}(W)}^{\zeta}_{\operatorname{\mathbf{d}}}}}{\mathbb{Q}}_{{\mathfrak{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}))\cong \lim_{{\mathbf{f}}\mapsto\infty}\left(\operatorname{\mathcal{H}}^n\left(\dim_!\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}\tilde{l}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}\tilde{l}^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}\right)\right)$$

For sufficiently large ${\mathbf{f}}$ we may write the left hand side of (\[ft1\]) as $$\operatorname{\mathcal{H}}^n\left(l_{\operatorname{\mathbf{d}},*}^{\zeta}l_{\operatorname{\mathbf{d}}}^{\zeta,*}\operatorname{\mathcal{H}}\left(\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes {\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}\right)\right)\cong\operatorname{\mathcal{H}}^n\left(l_{\operatorname{\mathbf{d}},*}^{\zeta}l_{\operatorname{\mathbf{d}}}^{\zeta,*}\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}\right)$$ where the isomorphism is an application of the decomposition theorem (Proposition \[basicfacts\](\[vanDec\])).  It is sufficient to observe that there is an natural isomorphism $$\label{swapper}
l^{\zeta}_{\operatorname{\mathbf{d}},*}l^{\zeta,*}_{\operatorname{\mathbf{d}}}\pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}\rightarrow \pi^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},!}l^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}l^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}\mathbb{Q}_{{\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}$$ by base change.  Similarly, we write the left hand side of (\[ft2\]) as $$\operatorname{\mathcal{H}}^n\left(\dim_!l_{\operatorname{\mathbf{d}},*}^{\zeta}l_{\operatorname{\mathbf{d}}}^{\zeta,*}\pi_{{\mathbf{f}},\operatorname{\mathbf{d}},!}^{\zeta}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}}{\mathbb{Q}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}\right)$$ and we obtain (\[ft2\]) by applying $\operatorname{\mathcal{H}}^n(\dim_!)$ to the isomorphism (\[swapper\]) for sufficiently large ${\mathbf{f}}$.

We define $$\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}={\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}}l^{\zeta,*}_{\mu}{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}}\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$$ and $$\operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\operatorname{\mathbf{d}}})^{\operatorname{Sp}}={\mathbb{D}}^{\operatorname{mon}}_{\Lambda_{\mu}^{\zeta}}(\lim_{{\mathbf{f}}\mapsto\infty}(\operatorname{H}_c({\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta},l_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta,*}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}}}}{\mathcal{IC}}_{{\mathcal{M}}_{{\mathbf{f}},\operatorname{\mathbf{d}}}^{\zeta}}({\mathbb{Q}})\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}).$$ As a word of explanation for the notation above, note that we do not claim that $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}$ is isomorphic to its total cohomology.  That is why we prefer this notation over, say $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\operatorname{Sp},\zeta}_{W,\mu})$. As a further word of explanation, note that when ${\mathcal{M}}^{\operatorname{Sp},\zeta}_{\mu}={\mathcal{M}}^{\zeta}_{\mu}$ the above definitions recover the definitions already provided in the absence of the superscript $\operatorname{Sp}$ by Proposition \[basicfacts\](\[monPD\]).  Applying the restriction functor $l^{\zeta,*}_{\mu}$ to Theorem \[ThmA\], we obtain the corollary:

\[SpWU\] Let $\zeta$ be a $\mu$-generic stability condition.  We define the restricted DT invariants $${\mathcal{DT}}^{\operatorname{Sp},\zeta}_{W,\operatorname{\mathbf{d}}}={\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta}_{\operatorname{\mathbf{d}}}}l^{\zeta,*}_{\operatorname{\mathbf{d}}}{\mathcal{DT}}^{\zeta}_{W,\operatorname{\mathbf{d}}}$$ and $$\operatorname{DT}^{\operatorname{Sp},\zeta}_{W,\operatorname{\mathbf{d}}}=\dim_*{\mathcal{DT}}^{\operatorname{Sp},\zeta}_{W,\operatorname{\mathbf{d}}}.$$ Then there are isomorphisms $$\label{fri}
\operatorname{Sym}_{\boxtimes_+}\left(\operatorname{DT}_{W,\mu}^{\operatorname{Sp},\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\cong \operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}$$ and $$\label{sri}
\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\operatorname{Sp},\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\cong\operatorname{\mathcal{H}}(\mathcal{A}_{W,\mu}^{\zeta})^{\operatorname{Sp}}$$ in ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}({\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}))$ and ${\mathcal{D}^{\geq, \mathrm{lf}}}(\operatorname{MMHM}(\Lambda^{\zeta}_{\mu}))$ respectively. 

We prove the existence of the isomorphism (\[sri\]), the isomorphism (\[fri\]) is then obtained by applying $\dim_*$. The isomorphism (\[sri\]) is given by the chain of isomorphisms $$\begin{aligned}
\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\operatorname{Sp},\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\cong 
& \operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}l^{\zeta,*}_{\mu}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)
\\
\cong &\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}\left(l^{\zeta,*}_{\mu}{\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}^{\vee}\right)\right)
\\ 
\cong&{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}l^{\zeta,*}_{\mu}\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})^{\vee}_{\operatorname{vir}}\right)\\
\cong &{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}l^{\zeta,*}_{\mu}\operatorname{Sym}_{\boxtimes_{\oplus}}\left({\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\\
\cong &{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}l^{\zeta,*}_{\mu}{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}\operatorname{Sym}_{\boxtimes_{\oplus}}\left(\left({\mathcal{DT}}_{W,\mu}^{\zeta}\otimes\operatorname{H}(\mathbb{C}\mathbb{P}^{\infty})_{\operatorname{vir}}\right)\right)\\
\cong &{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}l^{\zeta,*}_{\mu}{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}^{\zeta\operatorname{-ss}}_{\mu}}\operatorname{\mathcal{H}}\left(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta}\right).\end{aligned}$$

In the same way that Theorem \[strongPBW\] is an improvement upon Theorem \[ThmA\], we can improve upon Corollary \[SpWU\], and there is a cohomological wall crossing isomorphism incorporating also the ‘strong’ PBW theorem, which we finish by briefly describing. 

Let $\operatorname{\mathbf{d}}=\operatorname{\mathbf{d}}'+\operatorname{\mathbf{d}}''$ with $\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}$. We consider the maps $$\begin{aligned}
\alpha^{\operatorname{Sp},\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}=&{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}l^{\zeta,*}_{\mu}{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}\alpha^{\zeta}_{\mu}\\
\beta^{\operatorname{Sp},\zeta}_{\mu}=&{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\operatorname{\mathbf{d}}}^{\zeta\operatorname{-ss}}}l^{\zeta,*}_{\mu}{\mathbb{D}}^{\operatorname{mon}}_{{\mathcal{M}}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta\operatorname{-ss}}}\beta^{\zeta}_{\mu}\\
\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})^{\operatorname{Sp}}:= &\left(\beta^{\operatorname{Sp},\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\otimes {\mathbb{L}}^{-(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')/2}\right)\circ \left(\alpha^{\operatorname{Sp},\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}\otimes {\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2}\right)\\
\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})^{\operatorname{Sp}}:= &\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}}\operatorname{\mathcal{H}}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})^{\operatorname{Sp}}.\end{aligned}$$ Similarly, we define the natural isomorphism $$\begin{aligned}
\overline{\alpha}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\operatorname{Sp},\zeta}\colon&\lim_{N\mapsto \infty}\operatorname{\mathcal{H}}(\dim_*p_{\operatorname{\mathbf{d}}',N,*}l^{\zeta}_{\operatorname{\mathbf{d}}',N,*}l^{\zeta,*}_{\operatorname{\mathbf{d}}',N}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}',N}^{\zeta\operatorname{-ss}}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',N}^{\zeta}})\boxtimes \\&\lim_{N\mapsto \infty}\operatorname{\mathcal{H}}(\dim_*p_{\operatorname{\mathbf{d}}'',N,*}l^{\zeta}_{\operatorname{\mathbf{d}}'',N,*}l^{\zeta,*}_{\operatorname{\mathbf{d}}'',N}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}'',N}^{\zeta}})\rightarrow\\
&\operatorname{\mathcal{H}}(\dim_*p_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}l^{\zeta}_{N,*}l^{\zeta,*}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}{\phi^{\operatorname{mon}}_{{\mathcal{T}r(W)}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta}})\end{aligned}$$ and define

$$\overline{\beta}^{\operatorname{Sp},\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}=\lim_{N\mapsto\infty}\left(\operatorname{\mathcal{H}}\left(\dim_*p^{\zeta}_{\operatorname{\mathbf{d}},N,*}l^{\zeta}_{\operatorname{\mathbf{d}},N,*}l^{\zeta,*}_{\operatorname{\mathbf{d}},N}{\phi^{\operatorname{mon}}_{{\mathrm{Tr}(W)}_{\operatorname{\mathbf{d}},N}^{\zeta}}}(s^{\zeta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N,*}{\mathbb{Q}}_{X_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'',N}^{\zeta\operatorname{-ss}}}\otimes {\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}'')}\rightarrow {\mathbb{Q}}_{X_{\operatorname{\mathbf{d}},N}^{\zeta\operatorname{-ss}}})\right)\right)$$ and composing appropriate twists we obtain the maps $$\operatorname{H}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})^{\operatorname{Sp}}=(\overline{\beta}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\operatorname{Sp},\zeta}\otimes{\mathbb{L}}^{\langle \operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'\rangle/2})(\overline{\alpha}_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\operatorname{Sp},\zeta}\otimes{\mathbb{L}}^{(\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}')/2+(\operatorname{\mathbf{d}}'',\operatorname{\mathbf{d}}'')/2})$$ and $$\operatorname{H}(\operatorname{\tilde{\ast}}_{W,\mu}^{\zeta})^{\operatorname{Sp}}:=\bigoplus_{\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''\in\Lambda_{\mu}^{\zeta}}\operatorname{H}(\operatorname{\tilde{\ast}}_{W,\operatorname{\mathbf{d}}',\operatorname{\mathbf{d}}''}^{\zeta})^{\operatorname{Sp}}\colon \operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}\boxtimes_+^{\operatorname{tw}} \operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}\rightarrow \operatorname{H}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}.$$

We consider the filtration $$\operatorname{\mathfrak{P}}^p\left(\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},l^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}l^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\right)\right):=\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},l^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}l^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\tau_{\leq p}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}\right)$$ on $\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})^{\operatorname{Sp}}$. Then by definition, we have an isomorphism of algebras $$\label{restrictedAG}
\operatorname{\mathcal{H}}(\dim_*(\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})^{\operatorname{Sp}}))\cong\operatorname{Gr}_{\operatorname{\mathfrak{P}}}\left(\operatorname{H}(\operatorname{\mathcal{A}}_{W,\mu}^{\zeta})^{\operatorname{Sp}}\right).$$ Since the multiplication on $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})^{\operatorname{Sp}}$ is defined by restricting the multiplication on $\operatorname{\mathcal{H}}(\operatorname{\mathcal{A}}^{\zeta}_{W,\mu})$, we deduce that the modification of Theorem \[strongPBW\] obtained by adding the superscript $\operatorname{Sp}$ everywhere remains true, where all relative isomorphisms are obtained by restricting the isomorphisms in the statement of Theorem \[strongPBW\] to the locus ${\mathcal{M}}^{\operatorname{Sp}}$ or ${\mathcal{M}}^{\operatorname{Sp},\zeta\operatorname{-ss}}$. By (\[restrictedAG\]), the statements of Theorem \[strongPBW\] over the base $\mathbb{N}^{Q_0}$ follow as in the proof of Theorem \[strongPBW\], by observing that all required maps become isomorphisms after passing to the perverse filtration. 

For the purposes of trying to produce interesting Lie algebras of primitive elements, the strategy is to find ${\mathcal{M}}^{\operatorname{Sp},\zeta}_{\mu}$ such that the definition $$\operatorname{\mathfrak{P}}_{\operatorname{Sp}}^p\left(\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},l^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}l^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\right)\right):=\operatorname{H}\left({\mathcal{M}}^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}}},\tau_{\leq p}l^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}l^{\zeta,*}_{{\mathbf{f}},\operatorname{\mathbf{d}}}\pi ^{\zeta}_{{\mathbf{f}},\operatorname{\mathbf{d}},*}{\mathfrak{IC}}_{W,{\mathbf{f}},\operatorname{\mathbf{d}}}\otimes{\mathbb{L}}^{-{\mathbf{f}}\cdot \operatorname{\mathbf{d}}}\right)$$ provides an alternative filtration of $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})^{\operatorname{Sp}}$. In this case, the argument for the vanishing of the Lie bracket does not apply, and we obtain for each such filtration a Lie algebra $\mathfrak{g}_Q[u]$ of primitive elements with possibly nonzero Lie bracket, such that $\operatorname{H}(\mathcal{A}_{W,\mu}^{\zeta})^{\operatorname{Sp}}$ is a deformation of $U(\mathfrak{g}_Q[u])$.

<span style="font-variant:small-caps;">B. Davison: EPFL SB, B1 437 (Bâtiment MA), Station 8, CH-1015 Lausanne Switzerland</span>\
*E-mail address:* `nicholas.davison@epfl.ch`\
\
<span style="font-variant:small-caps;">S. Meinhardt: Fachbereich C, Bergische Universität Wuppertal, Gau[ß]{}stra[ß]{}e 20, 42119 Wuppertal, Germany</span>\
*E-mail address:* `meinhardt@uni-wuppertal.de`\

[^1]: Set $Y={\mathcal{M}}_{\mu}^{\zeta\operatorname{-ss}}$. Our definition differs a little from that in [@MSS11], in that they consider the direct image along $\pi\colon Y^n\rightarrow \operatorname{Sym}^n Y$, while we consider the pushforward to $Y$ along $\oplus$. However, the map $\oplus$ factors through $\pi$, and so our definition is the direct image of theirs.