<figure><img src="image_1.png" /></figure>

HAL
open science

# Bases of tensor products and geometric Satake correspondence

Pierre Baumann, Stéphane Gaussent, Peter Littelmann

<figure><img src="image_2.png" /></figure>

## ► To cite this version:

Pierre Baumann, Stéphane Gaussent, Peter Littelmann. Bases of tensor products and geometric Satake correspondence. Journal of the European Mathematical Society, 2022, 26 (3), pp.919-983. 10.4171/JEMS/1302 . hal-02928298

HAL Id: hal-02928298

https://hal.science/hal-02928298

Submitted on 29 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_3.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_1.png" /></figure>

HAL
open science

# Bases of tensor products and geometric Satake correspondence

Pierre Baumann, Stéphane Gaussent, Peter Littelmann

► To cite this version:

Pierre Baumann, Stéphane Gaussent, Peter Littelmann. Bases of tensor products and geometric Satake correspondence. Journal of the European Mathematical Society, 2022, 26 (3), pp.919-983. 10.4171/JEMS/1302 . hal-02928298

HAL Id: hal-02928298

https://hal.science/hal-02928298v1

Submitted on 29 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Pierre Baumann · Stéphane Gaussent · Peter Littelmann

# Bases of tensor products and geometric Satake correspondence

*To the memory of C. S. Seshadri*

Received September 30, 2020; revised March 18, 2021

**Abstract.** The geometric Satake correspondence can be regarded as a geometric construction of rational representations of a complex connected reductive group $G$. In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.

**Keywords.** Affine Grassmannian, Mirković–Vilonen cycles, tensor product, fusion product, canonical bases

## 1. Introduction

Let $G$ be a connected reductive group over the field of complex numbers, endowed with a Borel subgroup and a maximal torus. Let $\Lambda^+$ be the set of dominant weights relative to these data. Given $\lambda \in \Lambda^+$, denote by $V(\lambda)$ the irreducible rational representation of $G$ with highest weight $\lambda$. Given a finite sequence $\lambda = (\lambda_1, \dots, \lambda_n)$ in $\Lambda^+$, define

$$V(\lambda) = V(\lambda_1) \otimes \dots \otimes V(\lambda_n).$$

A construction due to Mirković and Vilonen [39] in the context of the geometric Satake correspondence endows the spaces $V(\lambda)$ with linear bases. Specifically, $V(\lambda)$ is identified with the intersection homology of a parabolic affine Schubert variety $Gr^{\lambda}$, while (after an ingenious use of hyperbolic localization) the fundamental classes of so-called “Mirković–

<footnote>Pierre Baumann: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS UMR 7501, 67084 Strasbourg, France; p.baumann@unistra.fr</footnote>

<footnote>Stéphane Gaussent: Institut Camille Jordan, Université de Lyon, UJM et CNRS UMR 5208, 42023 Saint-Étienne, France; stephane.gaussent@univ-st-etienne.fr</footnote>

<footnote>Peter Littelmann: Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany; peter.littelmann@math.uni-koeln.de</footnote>

<footnote>Mathematics Subject Classification (2020): Primary 22E46; Secondary 14M15</footnote>
Pierre Baumann · Stéphane Gaussent · Peter Littelmann

# Bases of tensor products and geometric Satake correspondence

To the memory of C. S. Seshadri

Received September 30, 2020; revised March 18, 2021

**Abstract.** The geometric Satake correspondence can be regarded as a geometric construction of rational representations of a complex connected reductive group $G$. In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.

**Keywords.** Affine Grassmannian, Mirković–Vilonen cycles, tensor product, fusion product, canonical bases

## 1. Introduction

Let $G$ be a connected reductive group over the field of complex numbers, endowed with a Borel subgroup and a maximal torus. Let $\Lambda^+$ be the set of dominant weights relative to these data. Given $\lambda \in \Lambda^+$, denote by $V(\lambda)$ the irreducible rational representation of $G$ with highest weight $\lambda$. Given a finite sequence $\lambda = (\lambda_1, \dots, \lambda_n)$ in $\Lambda^+$, define

$$V(\lambda) = V(\lambda_1) \otimes \dots \otimes V(\lambda_n).$$

A construction due to Mirković and Vilonen [39] in the context of the geometric Satake correspondence endows the spaces $V(\lambda)$ with linear bases. Specifically, $V(\lambda)$ is identified with the intersection homology of a parabolic affine Schubert variety $Gr^{\lambda}$, while (after an ingenious use of hyperbolic localization) the fundamental classes of so-called “Mirković–

<footnote>Pierre Baumann: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS UMR 7501, 67084 Strasbourg, France; p.baumann@unistra.fr</footnote>

<footnote>Stéphane Gaussent: Institut Camille Jordan, Université de Lyon, UJM et CNRS UMR 5208, 42023 Saint-Étienne, France; stephane.gaussent@univ-st-etienne.fr</footnote>

<footnote>Peter Littelmann: Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany; peter.littelmann@math.uni-koeln.de</footnote>

<footnote>Mathematics Subject Classification (2020): Primary 22E46; Secondary 14M15</footnote>
Vilenen cycles” contained in $\overline{\mathrm{Gr}}^{\lambda}$ form a basis of this intersection homology. In [22], Goncharov and Shen extend this construction to the tensor products $V(\lambda)$. In the present paper, we investigate the properties of these bases, which we call MV bases.

We show that the MV basis of a representation $V(\lambda)$ is compatible with the isotypic filtration of this representation. This basis is also compatible with the action of the Chevalley generators of the Lie algebra $\mathfrak{g}$ of $G$, in the sense that the leading terms of the action define on the basis the structure of a crystal in the sense of Kashiwara. (For $n=1$, this fact is due to Braverman and Gaitsgory [9].) Let us transport this crystal structure onto the set $\mathcal{L}(\lambda)$ of MV cycles, which naturally indexes the MV basis of $V(\lambda)$. Then, with the help of the path model [33], we prove that there is a natural crystal isomorphism

$$ \mathcal{L}(\lambda) \cong \mathcal{L}(\lambda_1) \otimes \cdots \otimes \mathcal{L}(\lambda_n). $$

(For $n=2$, this isomorphism is again due to Braverman and Gaitsgory.)

We study the transition matrix between the MV basis of a tensor product $V(\lambda)$ and the tensor product of the MV bases of the different factors $V(\lambda_1), \dots, V(\lambda_n)$. Using the fusion product in the sense of Beilinson and Drinfeld [6], we explain that the entries of this transition matrix can be computed as intersection multiplicities. As a consequence, the transition matrix is unitriangular with nonnegative integral entries.

The properties stated above are analogues of results obtained by Lusztig about the dual canonical bases. To be specific, Lusztig [37] defines a notion of based module (module endowed with a basis enjoying certain specific properties) over the quantized enveloping algebra $U_v(\mathfrak{g})$ and shows the following facts:

* A simple module over $U_v(\mathfrak{g})$, endowed with its canonical basis, is a based module.

* The tensor product of finitely many based modules can be endowed with a basis that makes it a based module. This basis is constructed from the tensor product of the bases of the factors by adding corrective terms in a unitriangular fashion.

* The basis of a based module is compatible with the decreasing isotypic filtration of the module. Each subquotient in this filtration, endowed with the induced basis, is isomorphic as a based module to the direct sum of copies of a simple module endowed with its canonical basis.

The dual canonical bases for the representations $V(\lambda)$ (see [14]) can then be defined by dualizing Lusztig's construction and specializing the quantum parameter at $v=1$.

Because of its compatibility with the isotypic filtration, the dual canonical basis of a tensor product $V(\lambda)$ yields a linear basis of the invariant subspace $V(\lambda)^G$. Just as well, the MV basis provides a linear basis (sometimes called the Satake basis) of $V(\lambda)^G$. The Satake basis and the dual canonical basis of $V(\lambda)^G$ generally differ (the paper [13] provides a counterexample); nonetheless we show that the Satake basis enjoys the first two items in Khovanov and Kuperberg's list of properties for the dual canonical basis [30]. In particular, after restriction to the fixed-point subspaces, the signed permutation

$$ V(\lambda_1) \otimes V(\lambda_2) \otimes \cdots \otimes V(\lambda_n) \xrightarrow{\approx} V(\lambda_2) \otimes \cdots \otimes V(\lambda_n) \otimes V(\lambda_1) $$

maps the Satake basis of the domain to that of the codomain.
Vilonen cycles” contained in $\overline{\mathrm{Gr}}^{\lambda}$ form a basis of this intersection homology. In [22], Goncharov and Shen extend this construction to the tensor products $V(\lambda)$. In the present paper, we investigate the properties of these bases, which we call MV bases.

We show that the MV basis of a representation $V(\lambda)$ is compatible with the isotypic filtration of this representation. This basis is also compatible with the action of the Chevalley generators of the Lie algebra $\mathfrak{g}$ of $G$, in the sense that the leading terms of the action define on the basis the structure of a crystal in the sense of Kashiwara. (For $n=1$, this fact is due to Braverman and Gaitsgory [9].) Let us transport this crystal structure onto the set $\mathcal{L}(\lambda)$ of MV cycles, which naturally indexes the MV basis of $V(\lambda)$. Then, with the help of the path model [33], we prove that there is a natural crystal isomorphism

$$ \mathcal{L}(\lambda) \cong \mathcal{L}(\lambda_1) \otimes \cdots \otimes \mathcal{L}(\lambda_n). $$

(For $n=2$, this isomorphism is again due to Braverman and Gaitsgory.)

We study the transition matrix between the MV basis of a tensor product $V(\lambda)$ and the tensor product of the MV bases of the different factors $V(\lambda_1), \dots, V(\lambda_n)$. Using the fusion product in the sense of Beilinson and Drinfeld [6], we explain that the entries of this transition matrix can be computed as intersection multiplicities. As a consequence, the transition matrix is unitriangular with nonnegative integral entries.

The properties stated above are analogues of results obtained by Lusztig about the dual canonical bases. To be specific, Lusztig [37] defines a notion of based module (module endowed with a basis enjoying certain specific properties) over the quantized enveloping algebra $U_v(\mathfrak{g})$ and shows the following facts:

* A simple module over $U_v(\mathfrak{g})$, endowed with its canonical basis, is a based module.

* The tensor product of finitely many based modules can be endowed with a basis that makes it a based module. This basis is constructed from the tensor product of the bases of the factors by adding corrective terms in a unitriangular fashion.

* The basis of a based module is compatible with the decreasing isotypic filtration of the module. Each subquotient in this filtration, endowed with the induced basis, is isomorphic as a based module to the direct sum of copies of a simple module endowed with its canonical basis.

The dual canonical bases for the representations $V(\lambda)$ (see [14]) can then be defined by dualizing Lusztig's construction and specializing the quantum parameter at $v=1$.

Because of its compatibility with the isotypic filtration, the dual canonical basis of a tensor product $V(\lambda)$ yields a linear basis of the invariant subspace $V(\lambda)^G$. Just as well, the MV basis provides a linear basis (sometimes called the Satake basis) of $V(\lambda)^G$. The Satake basis and the dual canonical basis of $V(\lambda)^G$ generally differ (the paper [13] provides a counterexample); nonetheless we show that the Satake basis enjoys the first two items in Khovanov and Kuperberg's list of properties for the dual canonical basis [30]. In particular, after restriction to the fixed-point subspaces, the signed permutation

$$ V(\lambda_1) \otimes V(\lambda_2) \otimes \cdots \otimes V(\lambda_n) \xrightarrow{\sim} V(\lambda_2) \otimes \cdots \otimes V(\lambda_n) \otimes V(\lambda_1) $$

maps the Satake basis of the domain to that of the codomain.
As explained in [3], the MV bases of the irreducible representations $V(\lambda)$ can be glued together to produce a basis of the algebra $\mathbb{C}[N]$ of regular functions on the unipotent radical $N$ of $B$. Of particular interest would be any relation to the cluster algebra structure on $\mathbb{C}[N]$ [7, 21]. The methods developed in the present paper allow for explicit computations. For instance, we show that the cluster monomials attached to certain seeds belong to the MV basis. However, $\mathbb{C}[N]$ is not of finite cluster type in general, which means that the cluster monomials do not span the whole space. We compute in type $D_4$ the MV basis element at a specific position not covered by cluster monomials; at this spot, the MV basis, the dual canonical basis and the dual semicanonical basis pairwise differ.

## 2. Mirković–Vilonen cycles and bases

In the whole paper $G$ is a connected reductive group over $\mathbb{C}$, endowed with a Borel subgroup $B$ and a maximal torus $T \subseteq B$. We denote by $\Lambda$ the character lattice of $T$, by $\Phi \subseteq \Lambda$ the root system of $(G, T)$, by $\Phi^\vee$ the coroot system, and by $W$ the Weyl group. The datum of $B$ determines a set of positive roots in $\Phi$. We denote the dominance order on $\Lambda$ by $\le$ and the cone of dominant weights by $\Lambda^+$. We denote the half-sum of the positive coroots by $\rho$ and regard it as a linear form $\rho: \Lambda \to \mathbb{Q}$; then $\rho(\alpha) = 1$ for each simple root $\alpha$. The Langlands dual of $G$ is the connected reductive group $G^\vee$ over $\mathbb{C}$ built from the dual torus $T^\vee = \Lambda \otimes_Z G_m$ and the root system $\Phi^\vee$. The positive coroots define a Borel subgroup $B^\vee \subseteq G^\vee$.

### 2.1. Recollection on the geometric Satake correspondence

The geometric Satake correspondence was devised by Lusztig [36] and given its definitive shape by Beilinson and Drinfeld [6] and Mirković and Vilonen [39]. Additional references for the material presented in this section are [46] and [4].

As recalled in the introduction of [41], loop groups appear under several guises across mathematics: there is the differential-geometric variant, the algebraic-geometric one, etc. We adopt the framework of Lie theory and Kac–Moody groups [31]. For instance, though the affine Grassmannian is a (generally nonreduced) ind-scheme, we will only look at its complex ind-variety structure.

Let $\mathcal{O} = \mathbb{C}[\llbracket z \rrbracket]$ be the ring of formal power series in $z$ with complex coefficients and let $\mathcal{K} = \mathbb{C}(\langle z \rangle)$ be the fraction field of $\mathcal{O}$. The affine Grassmannian of the Langlands dual $G^\vee$ is the homogeneous space $\mathrm{Gr} = G^\vee(\mathcal{K})/G^\vee(\mathcal{O})$. This space, a partial flag variety for an affine Kac–Moody group, is endowed with the structure of an ind-variety.

Each weight $\lambda \in \Lambda$ gives a point $z^\lambda$ in $T^\vee(\mathcal{K})$, whose image in $\mathrm{Gr}$ is denoted by $L_\lambda$. The $G^\vee(\mathcal{O})$-orbit through $L_\lambda$ in $\mathrm{Gr}$, denoted by $\mathrm{Gr}^\lambda$, is a smooth connected simply-connected variety of dimension $2\rho(\lambda)$. The Cartan decomposition implies that

$$ \mathrm{Gr} = \bigsqcup_{\lambda \in \Lambda^+} \mathrm{Gr}^\lambda; \quad \text{moreover} \quad \overline{\mathrm{Gr}^\lambda} = \bigsqcup_{\substack{\mu \in \Lambda^+ \\ \mu \le \lambda}} \mathrm{Gr}^\mu. $$
As explained in [3], the MV bases of the irreducible representations $V(\lambda)$ can be glued together to produce a basis of the algebra $\mathbb{C}[N]$ of regular functions on the unipotent radical $N$ of $B$. Of particular interest would be any relation to the cluster algebra structure on $\mathbb{C}[N]$ [7, 21]. The methods developed in the present paper allow for explicit computations. For instance, we show that the cluster monomials attached to certain seeds belong to the MV basis. However, $\mathbb{C}[N]$ is not of finite cluster type in general, which means that the cluster monomials do not span the whole space. We compute in type $D_4$ the MV basis element at a specific position not covered by cluster monomials; at this spot, the MV basis, the dual canonical basis and the dual semicanonical basis pairwise differ.

## 2. Mirković–Vilonen cycles and bases

In the whole paper $G$ is a connected reductive group over $\mathbb{C}$, endowed with a Borel subgroup $B$ and a maximal torus $T \subseteq B$. We denote by $\Lambda$ the character lattice of $T$, by $\Phi \subseteq \Lambda$ the root system of $(G, T)$, by $\Phi^\vee$ the coroot system, and by $W$ the Weyl group. The datum of $B$ determines a set of positive roots in $\Phi$. We denote the dominance order on $\Lambda$ by $\le$ and the cone of dominant weights by $\Lambda^+$. We denote the half-sum of the positive coroots by $\rho$ and regard it as a linear form $\rho: \Lambda \to \mathbb{Q}$; then $\rho(\alpha) = 1$ for each simple root $\alpha$. The Langlands dual of $G$ is the connected reductive group $G^\vee$ over $\mathbb{C}$ built from the dual torus $T^\vee = \Lambda \otimes_Z \mathbb{G}_m$ and the root system $\Phi^\vee$. The positive coroots define a Borel subgroup $B^\vee \subseteq G^\vee$.

### 2.1. Recollection on the geometric Satake correspondence

The geometric Satake correspondence was devised by Lusztig [36] and given its definitive shape by Beilinson and Drinfeld [6] and Mirković and Vilonen [39]. Additional references for the material presented in this section are [46] and [4].

As recalled in the introduction of [41], loop groups appear under several guises across mathematics: there is the differential-geometric variant, the algebraic-geometric one, etc. We adopt the framework of Lie theory and Kac–Moody groups [31]. For instance, though the affine Grassmannian is a (generally nonreduced) ind-scheme, we will only look at its complex ind-variety structure.

Let $\mathcal{O} = \mathbb{C}[\llbracket z \rrbracket]$ be the ring of formal power series in $z$ with complex coefficients and let $\mathcal{K} = \mathbb{C}(\langle z \rangle)$ be the fraction field of $\mathcal{O}$. The affine Grassmannian of the Langlands dual $G^\vee$ is the homogeneous space $\mathrm{Gr} = G^\vee(\mathcal{K})/G^\vee(\mathcal{O})$. This space, a partial flag variety for an affine Kac–Moody group, is endowed with the structure of an ind-variety.

Each weight $\lambda \in \Lambda$ gives a point $z^\lambda$ in $T^\vee(\mathcal{K})$, whose image in $\mathrm{Gr}$ is denoted by $L_\lambda$. The $G^\vee(\mathcal{O})$-orbit through $L_\lambda$ in $\mathrm{Gr}$, denoted by $\mathrm{Gr}^\lambda$, is a smooth connected simply-connected variety of dimension $2\rho(\lambda)$. The Cartan decomposition implies that

$$ \mathrm{Gr} = \bigsqcup_{\lambda \in \Lambda^+} \mathrm{Gr}^\lambda; \quad \text{moreover} \quad \overline{\mathrm{Gr}^\lambda} = \bigsqcup_{\substack{\mu \in \Lambda^+ \\ \mu \le \lambda}} \mathrm{Gr}^\mu. $$
Let Perv(Gr) be the category of $G^\vee(\mathcal{O})$-equivariant perverse sheaves on Gr (for the middle perversity) supported on finitely many $G^\vee(\mathcal{O})$-orbits, with coefficients in $\mathbb{C}$. This is an abelian semisimple category. The simple objects in Perv(Gr) are the intersection cohomology sheaves

$$ \mathcal{I}_{\lambda} = \mathrm{IC}(\overline{\mathrm{Gr}^{\lambda}}, \mathbb{C}). $$

(By convention, IC sheaves are shifted so as to be perverse.)

Let $\theta \in \Lambda$ be a dominant and regular weight. The embedding

$$ \mathbb{C}^{\times} \xrightarrow{\theta} T^{\vee}(\mathbb{C}) \to G^{\vee}(\mathcal{K}) $$

gives rise to an action of $\mathbb{C}^\times$ on Gr. For each $\mu \in \Lambda$, the point $L_\mu$ is fixed by this action; we denote its stable and unstable sets by

$$ S_{\mu} = \{x \in \mathrm{Gr} \mid \lim_{c \to 0} \theta(c) \cdot x = L_{\mu}\} \quad \text{and} \quad T_{\mu} = \{x \in \mathrm{Gr} \mid \lim_{c \to \infty} \theta(c) \cdot x = L_{\mu}\} $$

and denote the inclusion maps by $s_\mu : S_\mu \to \mathrm{Gr}$ and $t_\mu : T_\mu \to \mathrm{Gr}$.

Given $\mu \in \Lambda$ and $\mathcal{A} \in \mathrm{Perv}(\mathrm{Gr})$, Mirković and Vilonen [39, Theorem 3.5] identify the homology groups

$$ H_c(S_\mu, (s_\mu)^*\mathcal{A}) \quad \text{and} \quad H(T_\mu, (t_\mu)^!\mathcal{A}) $$

via Braden's hyperbolic localization, show that these groups are concentrated in degree $2\rho(\mu)$, and define

$$ F_{\mu}(\mathcal{A}) = H^{2\rho(\mu)}(T_{\mu}, (t_{\mu})^{!}\mathcal{A}) \quad \text{and} \quad F(\mathcal{A}) = \bigoplus_{\mu \in \Lambda} F_{\mu}(\mathcal{A}). $$

Then $F$ is an exact and faithful functor from $\mathrm{Perv}(\mathrm{Gr})$ to the category of finite-dimensional $\Lambda$-graded $\mathbb{C}$-vector spaces. Mirković and Vilonen prove that $F$ induces an equivalence $\bar{F}$ from $\mathrm{Perv}(\mathrm{Gr})$ to the category $\mathrm{Rep}(G)$ of finite-dimensional rational representations of $G$, the $\Lambda$-grading on $F(\mathcal{A})$ giving rise to the decomposition of $\bar{F}(\mathcal{A})$ into weight subspaces. In the course of the proof, it is shown that $\bar{F}$ maps $\mathcal{I}_{\lambda}$ to the irreducible highest weight representation $V(\lambda)$.

The map $G^\vee(\mathcal{K}) \to \mathrm{Gr}$ is a principal $G^\vee(\mathcal{O})$-bundle. From the $G^\vee(\mathcal{O})$-space $\mathrm{Gr}$, we form the associated bundle

$$ \mathrm{Gr}_2 = G^{\vee}(\mathcal{K}) \times_{G^{\vee}(\mathcal{O})} \mathrm{Gr}. $$

This space is called the 2-fold convolution variety and has the structure of an ind-variety. The action of $G^\vee(\mathcal{K})$ on $\mathrm{Gr}$ defines a map $m : \mathrm{Gr}_2 \to \mathrm{Gr}$ of ind-varieties. Let $p : G^\vee(\mathcal{K}) \to \mathrm{Gr}$ and $q : G^\vee(\mathcal{K}) \times \mathrm{Gr} \to \mathrm{Gr}_2$ be the quotient maps. Given two equivariant perverse sheaves $\mathcal{A}_1$ and $\mathcal{A}_2$ on $\mathrm{Gr}$, there is a unique equivariant perverse sheaf $\mathcal{A}_1 \boxtimes \mathcal{A}_2$ on $\mathrm{Gr}_2$ such that

$$ p^*\mathcal{A}_1 \boxtimes \mathcal{A}_2 = q^*(\mathcal{A}_1 \boxtimes \mathcal{A}_2) $$
Let Perv(Gr) be the category of $G^\vee(\mathcal{O})$-equivariant perverse sheaves on Gr (for the middle perversity) supported on finitely many $G^\vee(\mathcal{O})$-orbits, with coefficients in $\mathbb{C}$. This is an abelian semisimple category. The simple objects in Perv(Gr) are the intersection cohomology sheaves

$$ \mathcal{I}_{\lambda} = \mathrm{IC}(\overline{\mathrm{Gr}^{\lambda}}, \mathbb{C}). $$

(By convention, IC sheaves are shifted so as to be perverse.)

Let $\theta \in \Lambda$ be a dominant and regular weight. The embedding

$$ \mathbb{C}^{\times} \xrightarrow{\theta} T^{\vee}(\mathbb{C}) \to G^{\vee}(\mathcal{K}) $$

gives rise to an action of $\mathbb{C}^\times$ on Gr. For each $\mu \in \Lambda$, the point $L_\mu$ is fixed by this action; we denote its stable and unstable sets by

$$ S_{\mu} = \{x \in \mathrm{Gr} \mid \lim_{c \to 0} \theta(c) \cdot x = L_{\mu}\} \quad \text{and} \quad T_{\mu} = \{x \in \mathrm{Gr} \mid \lim_{c \to \infty} \theta(c) \cdot x = L_{\mu}\} $$

and denote the inclusion maps by $s_\mu : S_\mu \to \mathrm{Gr}$ and $t_\mu : T_\mu \to \mathrm{Gr}$.

Given $\mu \in \Lambda$ and $\mathcal{A} \in \mathrm{Perv}(\mathrm{Gr})$, Mirković and Vilonen [39, Theorem 3.5] identify the homology groups

$$ H_c(S_\mu, (s_\mu)^*\mathcal{A}) \quad \text{and} \quad H(T_\mu, (t_\mu)^!\mathcal{A}) $$

via Braden's hyperbolic localization, show that these groups are concentrated in degree $2\rho(\mu)$, and define

$$ F_{\mu}(\mathcal{A}) = H^{2\rho(\mu)}(T_{\mu}, (t_{\mu})^{!}\mathcal{A}) \quad \text{and} \quad F(\mathcal{A}) = \bigoplus_{\mu \in \Lambda} F_{\mu}(\mathcal{A}). $$

Then $F$ is an exact and faithful functor from Perv(Gr) to the category of finite-dimensional $\Lambda$-graded $\mathbb{C}$-vector spaces. Mirković and Vilonen prove that $F$ induces an equivalence $\bar{F}$ from Perv(Gr) to the category Rep($G$) of finite-dimensional rational representations of $G$, the $\Lambda$-grading on $F(\mathcal{A})$ giving rise to the decomposition of $\bar{F}(\mathcal{A})$ into weight subspaces. In the course of the proof, it is shown that $\bar{F}$ maps $\mathcal{I}_\lambda$ to the irreducible highest weight representation $V(\lambda)$.

The map $G^\vee(\mathcal{K}) \to \mathrm{Gr}$ is a principal $G^\vee(\mathcal{O})$-bundle. From the $G^\vee(\mathcal{O})$-space Gr, we form the associated bundle

$$ \mathrm{Gr}_2 = G^{\vee}(\mathcal{K}) \times^{G^{\vee}(\mathcal{O})} \mathrm{Gr}. $$

This space is called the 2-fold convolution variety and has the structure of an ind-variety. The action of $G^\vee(\mathcal{K})$ on Gr defines a map $m : \mathrm{Gr}_2 \to \mathrm{Gr}$ of ind-varieties. Let $p : G^\vee(\mathcal{K}) \to \mathrm{Gr}$ and $q : G^\vee(\mathcal{K}) \times \mathrm{Gr} \to \mathrm{Gr}_2$ be the quotient maps. Given two equivariant perverse sheaves $\mathcal{A}_1$ and $\mathcal{A}_2$ on Gr, there is a unique equivariant perverse sheaf $\mathcal{A}_1 \boxtimes \mathcal{A}_2$ on $\mathrm{Gr}_2$ such that

$$ p^*\mathcal{A}_1 \boxtimes \mathcal{A}_2 = q^*(\mathcal{A}_1 \boxtimes \mathcal{A}_2) $$
in the equivariant derived category of constructible sheaves on $G^{\vee}(\mathcal{K}) \times \text{Gr}$. We then define the convolution product of $\mathcal{A}_1$ and $\mathcal{A}_2$ to be

$$ \mathcal{A}_1 * \mathcal{A}_2 = m_*(\mathcal{A}_1 \boxtimes \mathcal{A}_2). $$

Using Beilinson and Drinfeld's fusion product, one defines associativity and commutativity constraints and obtains a monoidal structure on Perv(Gr). Then $F$ is a tensor functor; in particular, the fusion product imparts an explicit identification of $\Lambda$-graded vector spaces

$$ F(\mathcal{A}_1 * \mathcal{A}_2) \cong F(\mathcal{A}_1) \otimes F(\mathcal{A}_2) $$

for any $(\mathcal{A}_1, \mathcal{A}_2) \in \text{Perv}(\text{Gr})^2$.

## 2.2. Mirković-Vilonen cycles

In this paper we study tensor products $V(\lambda) = V(\lambda_1) \otimes \cdots \otimes V(\lambda_n)$, where $\lambda = (\lambda_1, \dots, \lambda_n)$ is a sequence of dominant weights. Accordingly, we want to consider convolution products

$$ \mathcal{I}_{\lambda} = \mathcal{I}_{\lambda_1} * \cdots * \mathcal{I}_{\lambda_n} $$

and therefore need the $n$-fold convolution variety

$$ \text{Gr}_n = \underbrace{G^{\vee}(\mathcal{K}) \times^{G^{\vee}(\Theta)} \dots \times^{G^{\vee}(\Theta)} G^{\vee}(\mathcal{K})}_{n \text{ factors } G^{\vee}(\mathcal{K})} / G^{\vee}(\Theta). $$

As is customary, we denote elements in $\text{Gr}_n$ as classes $[g_1, \dots, g_n]$ of tuples of elements in $G^{\vee}(\mathcal{K})$. Plainly $\text{Gr}_1 = \text{Gr}$; hence, we write the quotient map $G^{\vee}(\mathcal{K}) \to \text{Gr}$ as $g \mapsto [g]$. We define a map $m_n : \text{Gr}_n \to \text{Gr}$ by setting $m_n([g_1, \dots, g_n]) = [g_1 \dots g_n]$.

Given $G^{\vee}(\Theta)$-stable subsets $K_1, \dots, K_n$ of $\text{Gr}$, we define

$$ K_1 \tilde{\chi} \cdots \tilde{\chi} K_n = \{[g_1, \dots, g_n] \in \text{Gr}_n \mid [g_1] \in K_1, \dots, [g_n] \in K_n\}. $$

Alternatively, $K_1 \tilde{\chi} \cdots \tilde{\chi} K_n$ can be defined as

$$ \hat{K}_1 \times^{G^{\vee}(\Theta)} \dots \times^{G^{\vee}(\Theta)} \hat{K}_n / G^{\vee}(\Theta) $$

where each $\hat{K}_j$ is the preimage of $K_j$ under the quotient map $G^{\vee}(\mathcal{K}) \to \text{Gr}$.

For $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$, we set

$$ \text{Gr}_n^{\lambda} = \text{Gr}^{\lambda_1} \tilde{\chi} \cdots \tilde{\chi} \text{Gr}^{\lambda_n}. $$

Viewing $\text{Gr}_n^{\lambda}$ as an iterated fibration with base $\text{Gr}^{\lambda_1}$ and successive fibers $\text{Gr}^{\lambda_2}, \dots, \text{Gr}^{\lambda_n}$, we infer that it is a smooth connected simply-connected variety of dimension $2\rho(|\lambda|)$, where

$$ |\lambda| = \lambda_1 + \dots + \lambda_n. $$

Also

$$ \text{Gr}_n = \bigsqcup_{\lambda \in (\Lambda^+)^n} \text{Gr}_n^{\lambda} \quad \text{and} \quad \overline{\text{Gr}}_n^{\lambda} = \overline{\text{Gr}}^{\lambda_1} \tilde{\chi} \cdots \tilde{\chi} \overline{\text{Gr}}^{\lambda_n} = \bigsqcup_{\substack{\mu \in (\Lambda^+)^n \\ \mu_1 \le \lambda_1, \dots, \mu_n \le \lambda_n}} \text{Gr}_n^{\mu}. $$
in the equivariant derived category of constructible sheaves on $G^{\vee}(\mathcal{K}) \times \text{Gr}$. We then define the convolution product of $\mathcal{A}_1$ and $\mathcal{A}_2$ to be

$$ \mathcal{A}_1 * \mathcal{A}_2 = m_*(\mathcal{A}_1 \boxtimes \mathcal{A}_2). $$

Using Beilinson and Drinfeld's fusion product, one defines associativity and commutativity constraints and obtains a monoidal structure on Perv(Gr). Then $F$ is a tensor functor; in particular, the fusion product imparts an explicit identification of $\Lambda$-graded vector spaces

$$ F(\mathcal{A}_1 * \mathcal{A}_2) \cong F(\mathcal{A}_1) \otimes F(\mathcal{A}_2) $$

for any $(\mathcal{A}_1, \mathcal{A}_2) \in \text{Perv}(\text{Gr})^2$.

## 2.2. Mirković-Vilonen cycles

In this paper we study tensor products $V(\lambda) = V(\lambda_1) \otimes \cdots \otimes V(\lambda_n)$, where $\lambda = (\lambda_1, \dots, \lambda_n)$ is a sequence of dominant weights. Accordingly, we want to consider convolution products

$$ \mathcal{I}_{\lambda} = \mathcal{I}_{\lambda_1} * \cdots * \mathcal{I}_{\lambda_n} $$

and therefore need the $n$-fold convolution variety

$$ \text{Gr}_n = \underbrace{G^{\vee}(\mathcal{K}) \times^{G^{\vee}(\Theta)} \dots \times^{G^{\vee}(\Theta)} G^{\vee}(\mathcal{K})}_{n \text{ factors } G^{\vee}(\mathcal{K})} / G^{\vee}(\Theta). $$

As is customary, we denote elements in $\text{Gr}_n$ as classes $[g_1, \dots, g_n]$ of tuples of elements in $G^{\vee}(\mathcal{K})$. Plainly $\text{Gr}_1 = \text{Gr}$; hence, we write the quotient map $G^{\vee}(\mathcal{K}) \to \text{Gr}$ as $g \mapsto [g]$. We define a map $m_n : \text{Gr}_n \to \text{Gr}$ by setting $m_n([g_1, \dots, g_n]) = [g_1 \dots g_n]$.

Given $G^{\vee}(\Theta)$-stable subsets $K_1, \dots, K_n$ of $\text{Gr}$, we define

$$ K_1 \tilde{\chi} \cdots \tilde{\chi} K_n = \{[g_1, \dots, g_n] \in \text{Gr}_n \mid [g_1] \in K_1, \dots, [g_n] \in K_n\}. $$

Alternatively, $K_1 \tilde{\chi} \cdots \tilde{\chi} K_n$ can be defined as

$$ \hat{K}_1 \times^{G^{\vee}(\Theta)} \dots \times^{G^{\vee}(\Theta)} \hat{K}_n / G^{\vee}(\Theta) $$

where each $\hat{K}_j$ is the preimage of $K_j$ under the quotient map $G^{\vee}(\mathcal{K}) \to \text{Gr}$.

For $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$, we set

$$ \text{Gr}_{n}^{\lambda} = \text{Gr}^{\lambda_1} \tilde{\chi} \cdots \tilde{\chi} \text{Gr}^{\lambda_n}. $$

Viewing $\text{Gr}_n^\lambda$ as an iterated fibration with base $\text{Gr}^{\lambda_1}$ and successive fibers $\text{Gr}^{\lambda_2}, \dots, \text{Gr}^{\lambda_n}$, we infer that it is a smooth connected simply-connected variety of dimension $2\rho(|\lambda|)$, where

$$ |\lambda| = \lambda_1 + \dots + \lambda_n. $$

Also

$$ \text{Gr}_n = \bigsqcup_{\lambda \in (\Lambda^+)^n} \text{Gr}_n^\lambda \quad \text{and} \quad \overline{\text{Gr}}_n^\lambda = \overline{\text{Gr}^{\lambda_1}} \tilde{\chi} \cdots \tilde{\chi} \overline{\text{Gr}^{\lambda_n}} = \bigsqcup_{\substack{\mu \in (\Lambda^+)^n \\ \mu_1 \le \lambda_1, \dots, \mu_n \le \lambda_n}} \text{Gr}_n^\mu. $$
**Proposition 2.1.** Let $\lambda \in (\Lambda^+)^n$. Then $\mathcal{I}_\lambda$ is the direct image by $m_n$ of the intersection cohomology sheaf of $\overline{\mathrm{Gr}}_n^\lambda$ with trivial local system, to wit

$$ \mathcal{I}_{\lambda} = (m_n)_* \mathrm{IC}(\overline{\mathrm{Gr}}_n^{\lambda}, \mathbb{C}), $$

and the cohomology sheaves $\mathcal{H}^k \mathrm{IC}(\overline{\mathrm{Gr}}_n^\lambda, \mathbb{C})$ vanish unless $k$ and $2\rho(|\lambda|)$ have the same parity.

*Proof.* We content ourselves with the case $n=2$; the proof is the same in the general case but requires more notation. Working out the technicalities explained in [4, §1.16.4], we get

$$ \mathrm{IC}(\overline{\mathrm{Gr}}_2^{(\lambda_1, \lambda_2)}, \mathbb{C}) = \mathrm{IC}(\overline{\mathrm{Gr}}^{\lambda_1}, \mathbb{C}) \boxtimes \mathrm{IC}(\overline{\mathrm{Gr}}^{\lambda_2}, \mathbb{C}). $$

Applying $(m_2)_*$ then gives the desired equality, while the parity property follows from [36, Sect. 11]. ■

A key argument in Mirković and Vilonen's proof of the geometric Satake correspondence is the fact that for any $\lambda \in \Lambda^+$ and $\mu \in \Lambda$, all the irreducible components of $\overline{\mathrm{Gr}}^\lambda \cap S_\mu$ (respectively, $\overline{\mathrm{Gr}}^\lambda \cap T_\mu$) have dimension $\rho(\lambda + \mu)$ (respectively, $\rho(\lambda - \mu)$) (see [39, Theorem 3.2]). We need a similar result for the intersections $\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(S_\mu)$ and $\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\mu)$ inside the $n$-fold convolution variety.

Let $N^\vee$ be the unipotent radical of $B^\vee$. Then $S_\mu$ is the $N^\vee(\mathcal{K})$-orbit through $L_\mu$; this well-known fact follows from the easily proved inclusion $N^\vee(\mathcal{K})L_\mu \subseteq S_\mu$ and the Iwasawa decomposition

$$ G^\vee(\mathcal{K}) = \bigsqcup_{\mu \in \Lambda} N^\vee(\mathcal{K})z^\mu G^\vee(\Theta). \qquad (1) $$

We record that

$$ S_{\mu} = N^{\vee}(\mathcal{K})L_{\mu} = (N^{\vee}(\mathcal{K})z^{\mu}N^{\vee}(\Theta))/N^{\vee}(\Theta) = (N^{\vee}(\mathcal{K})z^{\mu})/N^{\vee}(\Theta) $$

and that for each $\lambda \in \Lambda^+$, the intersection $\overline{\mathrm{Gr}}^\lambda \cap S_\mu$ is stable under the action of the connected subgroup $N^\vee(\Theta)$, hence so is each irreducible component of this intersection.

The construction of the $n$-fold convolution variety is functorial in the group $G^\vee$. Applied to the inclusion $B^\vee \to G^\vee$, this remark provides a natural map

$$ \Psi: \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} (N^{\vee}(\mathcal{K})z^{\mu_1}) \times^{N^{\vee}(\Theta)} \dots \times^{N^{\vee}(\Theta)} (N^{\vee}(\mathcal{K})z^{\mu_n})/N^{\vee}(\Theta) \to \mathrm{Gr}_n. $$

Using (1), we easily see that $\Psi$ is bijective.

Given weights $\mu_1, \dots, \mu_n$ and $N^\vee(\Theta)$-stable subsets $Z_1 \subseteq S_{\mu_1}, \dots, Z_n \subseteq S_{\mu_n}$, we define

$$ Z_1 \times \dots \times Z_n = \tilde{Z}_1 \times^{N^\vee(\Theta)} \dots \times^{N^\vee(\Theta)} \tilde{Z}_n / N^\vee(\Theta) $$

where each $\tilde{Z}_j$ is the preimage of $Z_j$ under the quotient map $N^\vee(\mathcal{K})z^{\mu_j} \to S_{\mu_j}$. If $Z_1, \dots, Z_n$ are varieties, then $Z_1 \times \dots \times Z_n$ is an iterated fibration with base $Z_1$ and
**Proposition 2.1.** Let $\lambda \in (\Lambda^+)^n$. Then $\mathcal{I}_\lambda$ is the direct image by $m_n$ of the intersection cohomology sheaf of $\overline{\mathrm{Gr}}_n^\lambda$ with trivial local system, to wit

$$ \mathcal{I}_{\lambda} = (m_n)_* \mathrm{IC}(\overline{\mathrm{Gr}}_n^{\lambda}, \mathbb{C}), $$

and the cohomology sheaves $\mathcal{H}^k \mathrm{IC}(\overline{\mathrm{Gr}}_n^\lambda, \mathbb{C})$ vanish unless $k$ and $2\rho(|\lambda|)$ have the same parity.

*Proof.* We content ourselves with the case $n = 2$; the proof is the same in the general case but requires more notation. Working out the technicalities explained in [4, §1.16.4], we get

$$ \mathrm{IC}(\overline{\mathrm{Gr}}_2^{(\lambda_1, \lambda_2)}, \mathbb{C}) = \mathrm{IC}(\overline{\mathrm{Gr}}^{\lambda_1}, \mathbb{C}) \boxtimes \mathrm{IC}(\overline{\mathrm{Gr}}^{\lambda_2}, \mathbb{C}). $$

Applying $(m_2)_*$ then gives the desired equality, while the parity property follows from [36, Sect. 11]. ■

A key argument in Mirković and Vilonen's proof of the geometric Satake correspondence is the fact that for any $\lambda \in \Lambda^+$ and $\mu \in \Lambda$, all the irreducible components of $\overline{\mathrm{Gr}}^\lambda \cap S_\mu$ (respectively, $\overline{\mathrm{Gr}}^\lambda \cap T_\mu$) have dimension $\rho(\lambda + \mu)$ (respectively, $\rho(\lambda - \mu)$) (see [39, Theorem 3.2]). We need a similar result for the intersections $\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(S_\mu)$ and $\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\mu)$ inside the $n$-fold convolution variety.

Let $N^\vee$ be the unipotent radical of $B^\vee$. Then $S_\mu$ is the $N^\vee(\mathcal{K})$-orbit through $L_\mu$; this well-known fact follows from the easily proved inclusion $N^\vee(\mathcal{K})L_\mu \subseteq S_\mu$ and the Iwasawa decomposition

$$ G^\vee(\mathcal{K}) = \bigsqcup_{\mu \in \Lambda} N^\vee(\mathcal{K})z^\mu G^\vee(\Theta). \qquad (1) $$

We record that

$$ S_{\mu} = N^{\vee}(\mathcal{K})L_{\mu} = (N^{\vee}(\mathcal{K})z^{\mu}N^{\vee}(\Theta))/N^{\vee}(\Theta) = (N^{\vee}(\mathcal{K})z^{\mu})/N^{\vee}(\Theta) $$

and that for each $\lambda \in \Lambda^+$, the intersection $\overline{\mathrm{Gr}}^\lambda \cap S_\mu$ is stable under the action of the connected subgroup $N^\vee(\Theta)$, hence so is each irreducible component of this intersection.

The construction of the $n$-fold convolution variety is functorial in the group $G^\vee$. Applied to the inclusion $B^\vee \to G^\vee$, this remark provides a natural map

$$ \Psi: \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} (N^{\vee}(\mathcal{K})z^{\mu_1}) \times^{N^{\vee}(\Theta)} \dots \times^{N^{\vee}(\Theta)} (N^{\vee}(\mathcal{K})z^{\mu_n})/N^{\vee}(\Theta) \to \mathrm{Gr}_n. $$

Using (1), we easily see that $\Psi$ is bijective.

Given weights $\mu_1, \dots, \mu_n$ and $N^\vee(\Theta)$-stable subsets $Z_1 \subseteq S_{\mu_1}, \dots, Z_n \subseteq S_{\mu_n}$, we define

$$ Z_1 \times \dots \times Z_n = \tilde{Z}_1 \times^{N^\vee(\Theta)} \dots \times^{N^\vee(\Theta)} \tilde{Z}_n / N^\vee(\Theta) $$

where each $\tilde{Z}_j$ is the preimage of $Z_j$ under the quotient map $N^\vee(\mathcal{K})z^{\mu_j} \to S_{\mu_j}$. If $Z_1, \dots, Z_n$ are varieties, then $Z_1 \times \dots \times Z_n$ is an iterated fibration with base $Z_1$ and
successive fibers $Z_2, \dots, Z_n$ and $\Psi$ induces a homeomorphism from $Z_1 \times \cdots \times Z_n$ onto its image. The bijectivity of $\Psi$ implies that

$$ \mathrm{Gr}_n = \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} \Psi(S_{\mu_1} \times \cdots \times S_{\mu_n}). $$

We use $\mathrm{Irr}(-)$ to designate the set of irreducible components of its argument. For $\lambda \in \Lambda^+$, $\lambda \in (\Lambda^+)^n$, and $\mu \in \Lambda$, we define

$$ *\mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda \cap S_\mu) \quad \text{and} \quad *\mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(S_\mu)). $$

**Proposition 2.2.** Let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$ and let $\mu \in \Lambda$.

(i) All the irreducible components of $\overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(S_\mu)$ have dimension $\rho(|\lambda| + \mu)$.

(ii) The map $(Z_1, \dots, Z_n) \mapsto \overline{\Psi}(Z_1 \times \cdots \times Z_n)$ induces a bijection

$$ \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n \atop \mu_1 + \dots + \mu_n = \mu} *\mathcal{L}(\lambda_1)_{\mu_1} \times \cdots \times *\mathcal{L}(\lambda_n)_{\mu_n} \xrightarrow{\approx} *\mathcal{L}(\lambda)_{\mu}. $$

(The bar above $\Psi(Z_1 \times \cdots \times Z_n)$ means closure in $(m_n)^{-1}(S_\mu)$.)

*Proof.* One easily checks that $(m_n \circ \Psi)(S_{\mu_1} \times \cdots \times S_{\mu_n}) \subseteq S_{\mu_1+\dots+\mu_n}$, whence

$$ (m_n)^{-1}(S_\mu) = \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n \atop \mu_1 + \dots + \mu_n = \mu} \Psi(S_{\mu_1} \times \cdots \times S_{\mu_n}) $$

for any $\mu \in \Lambda$. Adding $\lambda = (\lambda_1, \dots, \lambda_n)$ to the mix, we see that

$$ \overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(S_\mu) = \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n \atop \mu_1 + \dots + \mu_n = \mu} \Psi(\overline{(\mathrm{Gr}^{\lambda_1}) \cap S_{\mu_1}}) \times \cdots \times \overline{(\mathrm{Gr}^{\lambda_n}) \cap S_{\mu_n}}) $$

is the disjoint union over $(\mu_1, \dots, \mu_n)$ of an iterated fibration with base $\overline{\mathrm{Gr}^{\lambda_1}} \cap S_{\mu_1}$ and successive fibers $\overline{\mathrm{Gr}^{\lambda_2}} \cap S_{\mu_2}, \dots, \overline{\mathrm{Gr}^{\lambda_n}} \cap S_{\mu_n}$. The proposition then follows from Mirković and Vilonen's dimension estimates. ■

For $\lambda \in \Lambda^+$, $\lambda \in (\Lambda^+)^n$, and $\mu \in \Lambda$, we similarly define

$$ \mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda \cap T_\mu) \quad \text{and} \quad \mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(T_\mu)). $$

Then all cycles in $\mathcal{L}(\lambda)_\mu$ have dimension $\rho(|\lambda| - \mu)$ and there is a natural bijection

$$ \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n \atop \mu_1 + \dots + \mu_n = \mu} \mathcal{L}(\lambda_1)_{\mu_1} \times \cdots \times \mathcal{L}(\lambda_n)_{\mu_n} \xrightarrow{\approx} \mathcal{L}(\lambda)_{\mu}. \qquad (2) $$

Elements in $*\mathcal{L}(\lambda)_\mu$, $\mathcal{L}(\lambda)_\mu$, $*\mathcal{L}(\lambda)_\mu$ or $\mathcal{L}(\lambda)_\mu$ are called Mirković-Vilonen (MV) cycles. For future use, we note that the map $Z \mapsto Z \cap \mathrm{Gr}_n^\lambda$ provides a bijection from $\mathcal{L}(\lambda)_\mu$ onto the set of irreducible components of $\mathrm{Gr}_n^\lambda \cap (m_n)^{-1}(T_\mu)$. (Each $Z \in \mathcal{L}(\lambda)_\mu$ meets the open subset $\mathrm{Gr}_n^\lambda$ of $\overline{\mathrm{Gr}_n^\lambda}$, because the dimension of $(\overline{\mathrm{Gr}_n^\lambda} \setminus \mathrm{Gr}_n^\lambda) \cap (m_n)^{-1}(T_\mu)$ is smaller than the dimension of $Z$.)
successive fibers $Z_2, \dots, Z_n$ and $\Psi$ induces a homeomorphism from $Z_1 \times \cdots \times Z_n$ onto its image. The bijectivity of $\Psi$ implies that

$$ \mathrm{Gr}_n = \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} \Psi(S_{\mu_1} \times \cdots \times S_{\mu_n}). $$

We use $\mathrm{Irr}(-)$ to designate the set of irreducible components of its argument. For $\lambda \in \Lambda^+$, $\lambda \in (\Lambda^+)^n$, and $\mu \in \Lambda$, we define

$$ *\mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda \cap S_\mu) \quad \text{and} \quad *\mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(S_\mu)). $$

**Proposition 2.2.** Let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$ and let $\mu \in \Lambda$.

(i) All the irreducible components of $\overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(S_\mu)$ have dimension $\rho(|\lambda| + \mu)$.

(ii) The map $(Z_1, \dots, Z_n) \mapsto \overline{\Psi}(Z_1 \times \cdots \times Z_n)$ induces a bijection

$$ \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} \mu_1 \times \cdots \times \mu_n \xrightarrow{\approx} \overline{\mathcal{L}(\lambda)}_\mu. $$

(The bar above $\Psi(Z_1 \times \cdots \times Z_n)$ means closure in $(m_n)^{-1}(S_\mu)$.)

*Proof.* One easily checks that $(m_n \circ \Psi)(S_{\mu_1} \times \cdots \times S_{\mu_n}) \subseteq S_{\mu_1+\cdots+\mu_n}$, whence

$$ (m_n)^{-1}(S_\mu) = \bigsqcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \Psi(S_{\mu_1} \times \dots \times S_{\mu_n}) $$

for any $\mu \in \Lambda$. Adding $\lambda = (\lambda_1, \dots, \lambda_n)$ to the mix, we see that

$$ \overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(S_\mu) = \bigsqcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \Psi(\overline{\mathrm{Gr}}^{\lambda_1} \cap S_{\mu_1}) \times \cdots \times \overline{\mathrm{Gr}}^{\lambda_n} \cap S_{\mu_n}) $$

is the disjoint union over $(\mu_1, \dots, \mu_n)$ of an iterated fibration with base $\overline{\mathrm{Gr}}^{\lambda_1} \cap S_{\mu_1}$ and successive fibers $\overline{\mathrm{Gr}}^{\lambda_2} \cap S_{\mu_2}, \dots, \overline{\mathrm{Gr}}^{\lambda_n} \cap S_{\mu_n}$. The proposition then follows from Mirković and Vilonen's dimension estimates. ■

For $\lambda \in \Lambda^+$, $\lambda \in (\Lambda^+)^n$, and $\mu \in \Lambda$, we similarly define

$$ \mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda \cap T_\mu) \quad \text{and} \quad \mathcal{L}(\lambda)_\mu = \mathrm{Irr}(\overline{\mathrm{Gr}}^\lambda_n \cap (m_n)^{-1}(T_\mu)). $$

Then all cycles in $\mathcal{L}(\lambda)_\mu$ have dimension $\rho(|\lambda| - \mu)$ and there is a natural bijection

$$ \bigsqcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times \mathcal{L}(\lambda_n)_{\mu_n} \xrightarrow{\approx} \mathcal{L}(\lambda)_{\mu}. \quad (2) $$

Elements in $*\mathcal{L}(\lambda)_\mu$, $\mathcal{L}(\lambda)_\mu$, $*\mathcal{L}(\lambda)_\mu$ or $\mathcal{L}(\lambda)_\mu$ are called Mirković-Vilonen (MV) cycles. For future use, we note that the map $Z \mapsto Z \cap \mathrm{Gr}_n^\lambda$ provides a bijection from $\mathcal{L}(\lambda)_\mu$ onto the set of irreducible components of $\mathrm{Gr}_n^\lambda \cap (m_n)^{-1}(T_\mu)$. (Each $Z \in \mathcal{L}(\lambda)_\mu$ meets the open subset $\mathrm{Gr}_n^\lambda$ of $\overline{\mathrm{Gr}}_n^\lambda$, because the dimension of $(\overline{\mathrm{Gr}}_n^\lambda \setminus \mathrm{Gr}_n^\lambda) \cap (m_n)^{-1}(T_\mu)$ is smaller than the dimension of $Z$.)
## 2.3. Mirković–Vilonen bases

Following Goncharov and Shen [22, Sect. 2.4], we now define the MV basis of the tensor product representation

$$V(\lambda) = F(\mathcal{I}_{\lambda}) = \bigoplus_{\mu \in \Lambda} F_{\mu}(\mathcal{I}_{\lambda}).$$

Let $\lambda \in (\Lambda^+)^n$ and let $\mu \in \Lambda$. By base change in the Cartesian square

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu}) \arrow[r, "f"] & \overline{\mathrm{Gr}}_{n}^{\lambda} \\
m_n \arrow[r, "m_n"] & m_n \\
T_{\mu} \arrow[r, "t_{\mu}", "\cong"] & \overline{\mathrm{Gr}}
\arrow[down, "\cong"'] & \arrow[down, "\cong"']
\end{tikzcd}$$

we compute

$$\begin{align*}
F_{\mu}(\mathcal{I}_{\lambda}) &= H^{2\rho(\mu)}(T_{\mu}, (t_{\mu})^{!}(m_{n}) * \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \mathbb{C})) \\
&= H^{2\rho(\mu)}(\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (m_{n})^{-1}(T_{\mu}), f^{!} \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \mathbb{C})).
\end{align*}$$

Let $j : \mathrm{Gr}_{n}^{\lambda} \to \overline{\mathrm{Gr}}_{n}^{\lambda}$ and $g : \mathrm{Gr}_{n}^{\lambda} \cap (m_{n})^{-1}(T_{\mu}) \to \mathrm{Gr}_{n}^{\lambda}$ be the inclusion maps. We can then look at the sequence of maps

$$\begin{align*}
F_{\mu}(\mathcal{I}_{\lambda}) &\to H^{2\rho(\mu)}(\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu}), f^! j_* j^* \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \mathbb{C})) \\
&= H^{2\rho(\mu)}(\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu}), g^! \mathbb{C}_{\overline{\mathrm{Gr}}_{n}^{\lambda}}[\dim \mathrm{Gr}_{n}^{\lambda}]) \\
&\xrightarrow{\cap[\mathrm{Gr}_{n}^{\lambda}]} H_{2\rho(|\lambda|-\mu)}^{\mathrm{BM}}(\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu})).
\end{align*}$$

Here the first two maps carry out the restriction to $\mathrm{Gr}_{n}^{\lambda}$ (technically, an adjunction followed by a base change) and the last map is the Alexander duality.<fnref n="1" />

We claim that these maps are isomorphisms. For the Alexander duality, this comes from the smoothness of $\mathrm{Gr}_{n}^{\lambda}$. For the restriction, consider a stratum $\mathrm{Gr}_{n}^{\eta} \subseteq \overline{\mathrm{Gr}}_{n}^{\lambda}$ with $\eta \neq \lambda$; denoting the inclusion map by $i$ and using the perversity condition, the parity property in Proposition 2.1, and the dimension estimate for $\overline{\mathrm{Gr}}_{n}^{\eta} \cap (m_n)^{-1}(T_\mu)$, one checks that

$$H^k(\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\mu), f^! i_* i^! \mathrm{IC}(\overline{\mathrm{Gr}}_n^\lambda, \mathbb{C}))$$

vanishes if $k < 2\rho(\mu) + 2$; therefore the stratum $\mathrm{Gr}_{n}^{\eta}$ does not contribute to $F_{\mu}(\mathcal{I}_{\lambda})$.

To sum up there is a natural isomorphism

$$F_{\mu}(\mathcal{I}_{\lambda}) \xrightarrow{\sim} H_{2\rho(|\lambda|-\mu)}^{\mathrm{BM}}(\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu})). \quad (3)$$

The irreducible components $\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_\mu)$ all have dimension $\rho(|\lambda| - \mu)$ and their fundamental classes provide a basis of the Borel–Moore homology group above. Gathering these bases for all weights $\mu \in \Lambda$ produces what we call the *MV basis* of $V(\lambda)$.

<footnote n="1">¹Specifically, the generalization presented in [15, Sect. 19.1, equation (3)] or in [24, Theorem IX.4.7].</footnote>
## 2.3. Mirković–Vilonen bases

Following Goncharov and Shen [22, Sect. 2.4], we now define the MV basis of the tensor product representation

$$V(\lambda) = F(\mathcal{I}_{\lambda}) = \bigoplus_{\mu \in \Lambda} F_{\mu}(\mathcal{I}_{\lambda}).$$

Let $\lambda \in (\Lambda^+)^n$ and let $\mu \in \Lambda$. By base change in the Cartesian square

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu}) \arrow[r, "f"] & \overline{\mathrm{Gr}}_{n}^{\lambda} \\
m_n \arrow[r, "m_n"] & m_n \\
T_{\mu} \arrow[r, "t_{\mu}", "\cong"] & T_{\mu} \arrow[r, "\cong"] & \overline{\mathrm{Gr}}_{n}
\end{tikzcd}$$

we compute

$$\begin{align*}
F_{\mu}(\mathcal{I}_{\lambda}) &= H^{2\rho(\mu)}(T_{\mu}, (t_{\mu})^{!}(m_{n}) * \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \mathbb{C})) \\
&= H^{2\rho(\mu)}(\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (m_{n})^{-1}(T_{\mu}), f^{!} \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \mathbb{C})).
\end{align*}$$

Let $j : \mathrm{Gr}_{n}^{\lambda} \to \overline{\mathrm{Gr}}_{n}^{\lambda}$ and $g : \mathrm{Gr}_{n}^{\lambda} \cap (m_{n})^{-1}(T_{\mu}) \to \mathrm{Gr}_{n}^{\lambda}$ be the inclusion maps. We can then look at the sequence of maps

$$\begin{align*}
F_{\mu}(\mathcal{I}_{\lambda}) &\to H^{2\rho(\mu)}(\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu}), f^! j_* j^* \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \mathbb{C})) \\
&= H^{2\rho(\mu)}(\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu}), g^! \mathbb{C}_{\overline{\mathrm{Gr}}_{n}^{\lambda}}[\dim \mathrm{Gr}_{n}^{\lambda}]) \\
&\xrightarrow{\cap[\mathrm{Gr}_{n}^{\lambda}]} H_{2\rho(|\lambda|-\mu)}^{\mathrm{BM}}(\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu})).
\end{align*}$$

Here the first two maps carry out the restriction to $\mathrm{Gr}_{n}^{\lambda}$ (technically, an adjunction followed by a base change) and the last map is the Alexander duality.<fnref n="1" />

We claim that these maps are isomorphisms. For the Alexander duality, this comes from the smoothness of $\mathrm{Gr}_{n}^{\lambda}$. For the restriction, consider a stratum $\mathrm{Gr}_{n}^{\eta} \subseteq \overline{\mathrm{Gr}}_{n}^{\lambda}$ with $\eta \neq \lambda$; denoting the inclusion map by $i$ and using the perversity condition, the parity property in Proposition 2.1, and the dimension estimate for $\overline{\mathrm{Gr}}_{n}^{\eta} \cap (m_n)^{-1}(T_\mu)$, one checks that

$$H^k(\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\mu), f^! i_* i^! \mathrm{IC}(\overline{\mathrm{Gr}}_n^\lambda, \mathbb{C}))$$

vanishes if $k < 2\rho(\mu) + 2$; therefore the stratum $\mathrm{Gr}_{n}^{\eta}$ does not contribute to $F_{\mu}(\mathcal{I}_{\lambda})$.

To sum up there is a natural isomorphism

$$F_{\mu}(\mathcal{I}_{\lambda}) \xrightarrow{\sim} H_{2\rho(|\lambda|-\mu)}^{\mathrm{BM}}(\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_{\mu})). \quad (3)$$

The irreducible components $\mathrm{Gr}_{n}^{\lambda} \cap (m_n)^{-1}(T_\mu)$ all have dimension $\rho(|\lambda| - \mu)$ and their fundamental classes provide a basis of the Borel–Moore homology group above. Gathering these bases for all weights $\mu \in \Lambda$ produces what we call the *MV basis* of $V(\lambda)$.

<footnote n="1">¹Specifically, the generalization presented in [15, Sect. 19.1, equation (3)] or in [24, Theorem IX.4.7].</footnote>
### 3. L-perfect bases

In this section we consider a general setup, which captures properties shared by both the MV bases and the dual canonical bases. As before, $G$ is a connected reductive group over $\mathbb{C}$ endowed with a Borel subgroup $B$ and a maximal torus $T \subseteq B$, $\Lambda$ is the character lattice of $T$, and $\Phi$ and $\Phi^\vee$ are the root and coroot systems of $(G, T)$. We denote by $\{\alpha_i | i \in I\}$ the set of simple roots defined by $B$ and by $\{\alpha_i^\vee | i \in I\}$ the set of simple coroots. Again, $\leq$ is the dominance order on $\Lambda$ and $\Lambda^+$ is the cone of dominant weights. We regard the Weyl group $W$ as a subgroup of $\mathrm{Aut}(\Lambda)$; for $i \in I$, we denote by $s_i$ the simple reflection along the root $\alpha_i$. When needed, we choose simple root vectors $e_i$ and $f_i$ of weights $\pm\alpha_i$ in the Lie algebra of $G$ in such a way that $[e_i, f_i] = -\alpha_i^\vee$.

#### 3.1. Seminormal crystals

We start by recalling the following definitions due to Kashiwara [28]. A seminormal crystal is a set $B$ endowed with a map $\mathrm{wt}: B \to \Lambda$ and, for each “color” $i \in I$, a partition into a collection of finite oriented strings. This latter structure is recorded by the datum of operators

$$ \tilde{e}_i : B \to B \sqcup \{0\} \quad \text{and} \quad \tilde{f}_i : B \to B \sqcup \{0\} $$

which move an element of $b \in B$ upwards and downwards, respectively, along the string of color $i$ to which $b$ belongs. The special value 0 is assigned to $\tilde{e}_i(b)$ or $\tilde{f}_i(b)$ when $b$ is at the upper or lower end of a string of color $i$. For convenience one usually further sets $\tilde{e}_i(0) = \tilde{f}_i(0) = 0$. The position of $b$ in its string of color $i$ is recorded by functions $\varepsilon_i$ and $\varphi_i$ which are defined as follows:

$$ \varepsilon_i(b) = \max \{n \ge 0 \mid \tilde{e}_i(b) \ne 0\}, \quad \varphi_i(b) = \max \{n \ge 0 \mid \tilde{f}_i(b) \ne 0\}. $$

Two compatibility conditions between the weight map $\mathrm{wt}$ and the datum of the partitions into oriented strings are required: first, $\mathrm{wt}(b)$ increases by $\alpha_i$ when $b$ moves upwards the string of color $i$ to which it belongs; second, $\varphi_i(b) - \varepsilon_i(b) = \langle \alpha_i^\vee, \mathrm{wt}(b) \rangle$ for any $b \in B$ and any $i \in I$. These conditions imply that the image of a string of color $i$ by the map $\mathrm{wt}$ is stable under the action of the simple reflection $s_i$. As a consequence, the set $\{\mathrm{wt}(b) | b \in B\}$ is stable under the action of the Weyl group $W$.

The direct sum of two seminormal crystals $B_1$ and $B_2$ is defined to be just the disjoint union of the underlying sets. The tensor product $B_1 \otimes B_2$ is defined to be the Cartesian product of the sets endowed with the maps given in [28, §7.3]. Notably, for each color, the strings in $B_1 \otimes B_2$ are created from the strings contained in $B_1$ and in $B_2$ by the process illustrated by the picture below.

<figure><img src="image_4.png" /></figure>
### 3. L-perfect bases

In this section we consider a general setup, which captures properties shared by both the MV bases and the dual canonical bases. As before, $G$ is a connected reductive group over $\mathbb{C}$ endowed with a Borel subgroup $B$ and a maximal torus $T \subseteq B$, $\Lambda$ is the character lattice of $T$, and $\Phi$ and $\Phi^\vee$ are the root and coroot systems of $(G, T)$. We denote by $\{\alpha_i | i \in I\}$ the set of simple roots defined by $B$ and by $\{\alpha_i^\vee | i \in I\}$ the set of simple coroots. Again, $\leq$ is the dominance order on $\Lambda$ and $\Lambda^+$ is the cone of dominant weights. We regard the Weyl group $W$ as a subgroup of $\mathrm{Aut}(\Lambda)$; for $i \in I$, we denote by $s_i$ the simple reflection along the root $\alpha_i$. When needed, we choose simple root vectors $e_i$ and $f_i$ of weights $\pm\alpha_i$ in the Lie algebra of $G$ in such a way that $[e_i, f_i] = -\alpha_i^\vee$.

#### 3.1. Seminormal crystals

We start by recalling the following definitions due to Kashiwara [28]. A seminormal crystal is a set $B$ endowed with a map $\mathrm{wt}: B \to \Lambda$ and, for each “color” $i \in I$, a partition into a collection of finite oriented strings. This latter structure is recorded by the datum of operators

$$ \tilde{e}_i : B \to B \sqcup \{0\} \quad \text{and} \quad \tilde{f}_i : B \to B \sqcup \{0\} $$

which move an element of $b \in B$ upwards and downwards, respectively, along the string of color $i$ to which $b$ belongs. The special value 0 is assigned to $\tilde{e}_i(b)$ or $\tilde{f}_i(b)$ when $b$ is at the upper or lower end of a string of color $i$. For convenience one usually further sets $\tilde{e}_i(0) = \tilde{f}_i(0) = 0$. The position of $b$ in its string of color $i$ is recorded by functions $\varepsilon_i$ and $\varphi_i$ which are defined as follows:

$$ \varepsilon_i(b) = \max \{n \geq 0 \mid \tilde{e}_i(b) \neq 0\}, \quad \varphi_i(b) = \max \{n \geq 0 \mid \tilde{f}_i(b) \neq 0\}. $$

Two compatibility conditions between the weight map $\mathrm{wt}$ and the datum of the partitions into oriented strings are required: first, $\mathrm{wt}(b)$ increases by $\alpha_i$ when $b$ moves upwards the string of color $i$ to which it belongs; second, $\varphi_i(b) - \varepsilon_i(b) = \langle \alpha_i^\vee, \mathrm{wt}(b) \rangle$ for any $b \in B$ and any $i \in I$. These conditions imply that the image of a string of color $i$ by the map $\mathrm{wt}$ is stable under the action of the simple reflection $s_i$. As a consequence, the set $\{\mathrm{wt}(b) \mid b \in B\}$ is stable under the action of the Weyl group $W$.

The direct sum of two seminormal crystals $B_1$ and $B_2$ is defined to be just the disjoint union of the underlying sets. The tensor product $B_1 \otimes B_2$ is defined to be the Cartesian product of the sets endowed with the maps given in [28, §7.3]. Notably, for each color, the strings in $B_1 \otimes B_2$ are created from the strings contained in $B_1$ and in $B_2$ by the process illustrated by the picture below.

<figure><img src="image_3.png" /></figure>
A morphism $f: B \to C$ between two seminormal crystals is a map that preserves the weight and that commutes with all the operators $\tilde{e}_i$ and $\tilde{f}_i$. (In contrast with the more general definition given in [26], morphisms between seminormal crystals are necessarily strict.)

## 3.2. L-perfect bases

To a subset $J \subseteq I$ we attach the standard Levi subgroup $M_J$ of $G$, the cone

$$ \Lambda_J^+ = \{\lambda \in \Lambda \mid \forall j \in J, \langle \alpha_j^\vee, \lambda \rangle \ge 0\} $$

of dominant weights for $M_J$, and the $J$-dominance order $\le_J$ on $\Lambda$ defined by

$$ \mu \le_J \lambda \iff \lambda - \mu \in \operatorname{span}_{\mathbb{N}} \{\alpha_j \mid j \in J\}. $$

Given $\lambda \in \Lambda_J^+$ we denote by $V_J(\lambda)$ the irreducible rational representation of $M_J$ with highest weight $\lambda$. Given a finite sequence $\lambda = (\lambda_1, \dots, \lambda_n)$ in $\Lambda_J^+$ we define

$$ V_J(\lambda) = V_J(\lambda_1) \otimes \dots \otimes V_J(\lambda_n). $$

For $J = I$ we recover the conventions previously used by dropping the decoration $J$ in the notation $\Lambda_J^+$, $\le_J$ or $V_J(\lambda)$.

Let $V$ be a rational representation of $G$. With respect to the action of $M_J$ the space $V$ can be uniquely written as a direct sum of isotypic components

$$ V = \bigoplus_{\mu \in \Lambda_J^+} V_{J,\mu} $$

where $V_{J,\mu}$ is the sum of all subrepresentations of $\mathrm{res}_{M_J}^G(V)$ isomorphic to $V_J(\mu)$. We define

$$ V_{J, \le_J \mu} = \bigoplus_{\substack{v \in \Lambda_J^+ \\ v \le_J \mu}} V_{J, v}. $$

We say that a linear basis $B$ of $V$ is $L$-perfect<fnref n="2" /> if for each $J \subseteq I$ and each $\mu \in \Lambda_J^+$, (P1) the subspace $V_{J, \le_J \mu}$ is spanned by a subset of $B$;

(P2) the induced basis on the quotient $V_{J, \le_J \mu} / V_{J, <_J \mu} \cong V_{J, \mu}$ is compatible with a decomposition of the isotypic component as a direct sum of irreducible representations.

Taking $J = \emptyset$, we see that an $L$-perfect basis of $V$ consists of weight vectors (note that $\le_J$ is the trivial order on $\Lambda$). Now let $i \in I$, and for each nonnegative integer $\ell$, define

$$ V_{\{i\}, \le \ell} = \bigoplus_{\substack{\mu \in \Lambda \\ 0 \le \langle \alpha_i^\vee, \mu \rangle \le \ell}} V_{\{i\}, \mu}, $$

<footnote n="2">²$L$ stands for Levi. This notion of $L$-perfect basis appears unnamed (and in a dual form) in Braverman and Gaitsgory's paper [9, Sect. 4.3].</footnote>
A morphism $f: B \to C$ between two seminormal crystals is a map that preserves the weight and that commutes with all the operators $\tilde{e}_i$ and $\tilde{f}_i$. (In contrast with the more general definition given in [26], morphisms between seminormal crystals are necessarily strict.)

## 3.2. L-perfect bases

To a subset $J \subseteq I$ we attach the standard Levi subgroup $M_J$ of $G$, the cone

$$ \Lambda_J^+ = \{\lambda \in \Lambda \mid \forall j \in J, \langle \alpha_j^\vee, \lambda \rangle \ge 0\} $$

of dominant weights for $M_J$, and the $J$-dominance order $\le_J$ on $\Lambda$ defined by

$$ \mu \le_J \lambda \iff \lambda - \mu \in \operatorname{span}_{\mathbb{N}} \{\alpha_j \mid j \in J\}. $$

Given $\lambda \in \Lambda_J^+$ we denote by $V_J(\lambda)$ the irreducible rational representation of $M_J$ with highest weight $\lambda$. Given a finite sequence $\lambda = (\lambda_1, \dots, \lambda_n)$ in $\Lambda_J^+$ we define

$$ V_J(\lambda) = V_J(\lambda_1) \otimes \dots \otimes V_J(\lambda_n). $$

For $J = I$ we recover the conventions previously used by dropping the decoration $J$ in the notation $\Lambda_J^+$, $\le_J$ or $V_J(\lambda)$.

Let $V$ be a rational representation of $G$. With respect to the action of $M_J$ the space $V$ can be uniquely written as a direct sum of isotypic components

$$ V = \bigoplus_{\mu \in \Lambda_J^+} V_{J,\mu} $$

where $V_{J,\mu}$ is the sum of all subrepresentations of $\mathrm{res}_{M_J}^G(V)$ isomorphic to $V_J(\mu)$. We define

$$ V_{J, \le_J \mu} = \bigoplus_{\substack{v \in \Lambda_J^+ \\ v \le_J \mu}} V_{J, v}. $$

We say that a linear basis $B$ of $V$ is $L$-perfect<fnref n="2" /> if for each $J \subseteq I$ and each $\mu \in \Lambda_J^+$, (P1) the subspace $V_{J, \le_J \mu}$ is spanned by a subset of $B$;

(P2) the induced basis on the quotient $V_{J, \le_J \mu} / V_{J, <_J \mu} \cong V_{J, \mu}$ is compatible with a decomposition of the isotypic component as a direct sum of irreducible representations.

Taking $J = \emptyset$, we see that an $L$-perfect basis of $V$ consists of weight vectors (note that $\le_J$ is the trivial order on $\Lambda$). Now let $i \in I$, and for each nonnegative integer $\ell$, define

$$ V_{\{i\}, \le \ell} = \bigoplus_{\substack{\mu \in \Lambda \\ 0 \le \langle \alpha_i^\vee, \mu \rangle \le \ell}} V_{\{i\}, \mu}, $$

<footnote n="2">²$L$ stands for Levi. This notion of $L$-perfect basis appears unnamed (and in a dual form) in Braverman and Gaitsgory's paper [9, Sect. 4.3].</footnote>
the sum of all irreducible subrepresentations of $\text{res}_{M_{\{i\}}}^G(V)$ of dimension at most $\ell + 1$. If $B$ satisfies conditions (P1) and (P2) for $J = \{i\}$, then $V_{\{i\}, \le \ell}$ is spanned by $B \cap V_{\{i\}, \le \ell}$ and the induced basis on $V_{\{i\}, \le \ell} / V_{\{i\}, \le \ell-1}$ is compatible with a decomposition as a direct sum of irreducible representations. Therefore $(B \cap V_{\{i\}, \le \ell}) \setminus (B \cap V_{\{i\}, \le \ell-1})$ decomposes as the disjoint union of oriented strings of length $\ell$, in such a way that the simple root vector $e_i$ or $f_i$ acts on a basis vector of $V_{\{i\}, \le \ell} / V_{\{i\}, \le \ell-1}$ by moving it upwards or downwards along the string that contains it, up to a scalar.

We can sum up the discussion in the previous paragraph as follows: if $B$ satisfies (P1) and (P2) for all $J$ of cardinality $\le 1$, then $B$ is endowed with the structure of crystal and is perfect in the sense of Berenstein and Kazhdan [8, Definition 5.30].

**Lemma 3.1.** Let $B$ be an $L$-perfect basis of a rational representation $V$ of $G$ and let $B' \subseteq B$. Assume that the linear space $V'$ spanned by $B'$ is a subrepresentation of $V$. Then $B'$ is an $L$-perfect basis of $V'$ and (the image of) $B \setminus B'$ is an $L$-perfect basis of the quotient $V/V'$.

*Proof.* Let $J \subseteq I$ and $\mu \in \Lambda_J^+$. Then

$$V'_{J,\mu} = V' \cap V_{J,\mu} \quad \text{and} \quad V'_{J,\le J,\mu} = V' \cap V_{J,\le J,\mu}.$$

Now both spaces $V'$ and $V_{J, \le J, \mu}$ are spanned by a subset of the basis $B$, so their intersection $V'_{J, \le J, \mu}$ is spanned by a subset of $B$, namely $B' \cap V_{J, \le J, \mu}$.

Let $C$ be the image of $(B \cap V_{J, \le J, \mu}) \setminus (B \cap V_{J, < J, \mu})$ in the quotient $V_{J, \le J, \mu} / V_{J, < J, \mu} \cong V_{J, \mu}$. Then $C$ can be viewed as a basis of $V_{J, \mu}$ and it can be partitioned into disjoint subsets $C_1, \dots, C_n$ so that each $C_k$ spans an irreducible subrepresentation. By construction, the subspace $V'_{J, \mu}$ is spanned by a subset $C' \subseteq C$. Each subset $C_k$ can be either contained in $C'$ or disjoint from $C'$, depending on whether the subrepresentation that it spans is contained in $V'_{J, \mu}$ or meets $V'_{J, \mu}$ trivially. Therefore $C'$ is the disjoint union of some $C_k$, which means that $C'$ is compatible with a decomposition of $V'_{J, \mu}$ as a direct sum of irreducible subrepresentations.

Thus, $B'$ satisfies both conditions (P1) and (P2), and is therefore an $L$-perfect basis of $V'$. The proof that $B \setminus B'$ yields an $L$-perfect basis of the quotient $V/V'$ rests on similar arguments and is left to the reader. $\blacksquare$

Under the assumptions of Lemma 3.1, the subset $B'$ is a subcrystal of $B$. In other words, the crystal structure on $B$ is the direct sum of the crystal structures on $B'$ and $B \setminus B'$.

**Proposition 3.2.** Let $V$ be a rational representation of $G$. Up to isomorphism, the crystal of an $L$-perfect basis of $V$ depends only on $V$, and not on the basis.

*Proof.* Let $B$ be an $L$-perfect basis of $V$. The conditions imposed on $B$ with the choice $J = I$ imply the existence in $V$ of a composition series compatible with $B$. By Lemma 3.1, the crystal $B$ is the direct sum of the crystals of the $L$-perfect bases induced by $B$ on the subquotients. It thus suffices to prove the desired uniqueness property in the particular case where $V$ is an irreducible representation, which in fact is just Theorem 5.37 in [8]. $\blacksquare$
the sum of all irreducible subrepresentations of $\text{res}_{M_{\langle i \rangle}}^G(V)$ of dimension at most $\ell + 1$. If $B$ satisfies conditions (P1) and (P2) for $J = \{i\}$, then $V_{\{i\}, \le \ell}$ is spanned by $B \cap V_{\{i\}, \le \ell}$ and the induced basis on $V_{\{i\}, \le \ell} / V_{\{i\}, \le \ell-1}$ is compatible with a decomposition as a direct sum of irreducible representations. Therefore $(B \cap V_{\{i\}, \le \ell}) \setminus (B \cap V_{\{i\}, \le \ell-1})$ decomposes as the disjoint union of oriented strings of length $\ell$, in such a way that the simple root vector $e_i$ or $f_i$ acts on a basis vector of $V_{\{i\}, \le \ell} / V_{\{i\}, \le \ell-1}$ by moving it upwards or downwards along the string that contains it, up to a scalar.

We can sum up the discussion in the previous paragraph as follows: if $B$ satisfies (P1) and (P2) for all $J$ of cardinality $\le 1$, then $B$ is endowed with the structure of crystal and is perfect in the sense of Berenstein and Kazhdan [8, Definition 5.30].

**Lemma 3.1.** Let $B$ be an $L$-perfect basis of a rational representation $V$ of $G$ and let $B' \subseteq B$. Assume that the linear space $V'$ spanned by $B'$ is a subrepresentation of $V$. Then $B'$ is an $L$-perfect basis of $V'$ and (the image of) $B \setminus B'$ is an $L$-perfect basis of the quotient $V/V'$.

*Proof.* Let $J \subseteq I$ and $\mu \in \Lambda_J^+$. Then

$$V'_{J,\mu} = V' \cap V_{J,\mu} \quad \text{and} \quad V'_{J,\le J,\mu} = V' \cap V_{J,\le J,\mu}.$$

Now both spaces $V'$ and $V_{J, \le J, \mu}$ are spanned by a subset of the basis $B$, so their intersection $V'_{J, \le J, \mu}$ is spanned by a subset of $B$, namely $B' \cap V_{J, \le J, \mu}$.

Let $C$ be the image of $(B \cap V_{J, \le J, \mu}) \setminus (B \cap V_{J, < J, \mu})$ in the quotient $V_{J, \le J, \mu} / V_{J, < J, \mu} \cong V_{J, \mu}$. Then $C$ can be viewed as a basis of $V_{J, \mu}$ and it can be partitioned into disjoint subsets $C_1, \dots, C_n$ so that each $C_k$ spans an irreducible subrepresentation. By construction, the subspace $V'_{J, \mu}$ is spanned by a subset $C' \subseteq C$. Each subset $C_k$ can be either contained in $C'$ or disjoint from $C'$, depending on whether the subrepresentation that it spans is contained in $V'_{J, \mu}$ or meets $V'_{J, \mu}$ trivially. Therefore $C'$ is the disjoint union of some $C_k$, which means that $C'$ is compatible with a decomposition of $V'_{J, \mu}$ as a direct sum of irreducible subrepresentations.

Thus, $B'$ satisfies both conditions (P1) and (P2), and is therefore an $L$-perfect basis of $V'$. The proof that $B \setminus B'$ yields an $L$-perfect basis of the quotient $V/V'$ rests on similar arguments and is left to the reader. $\blacksquare$

Under the assumptions of Lemma 3.1, the subset $B'$ is a subcrystal of $B$. In other words, the crystal structure on $B$ is the direct sum of the crystal structures on $B'$ and $B \setminus B'$.

**Proposition 3.2.** Let $V$ be a rational representation of $G$. Up to isomorphism, the crystal of an $L$-perfect basis of $V$ depends only on $V$, and not on the basis.

*Proof.* Let $B$ be an $L$-perfect basis of $V$. The conditions imposed on $B$ with the choice $J = I$ imply the existence in $V$ of a composition series compatible with $B$. By Lemma 3.1, the crystal $B$ is the direct sum of the crystals of the $L$-perfect bases induced by $B$ on the subquotients. It thus suffices to prove the desired uniqueness property in the particular case where $V$ is an irreducible representation, which in fact is just Theorem 5.37 in [8]. $\blacksquare$
In particular, the crystal of an L-perfect basis of an irreducible representation $V(\lambda)$ is unique. We use henceforth the notation $B(\lambda)$ for the associated crystal.

**Remark 3.3.** The crystal $B(\lambda)$ of an irreducible representation $V(\lambda)$ was introduced by Kashiwara in the context of representations of quantum groups. The definition via crystallization at $q = 0$ and the definition via the combinatorics of L-perfect bases yield the same crystal; this follows from [27, Sect. 5].

Fortunately, this nice little theory is not empty. As mentioned in the introduction, any tensor product of irreducible representations has an L-perfect basis, namely its dual canonical basis. Another example of an L-perfect basis: it can be shown, in the case where $G$ is simply laced, that the dual semicanonical basis of an irreducible representation is L-perfect.

**Theorem 3.4.** *The MV basis of a tensor product of irreducible representations is L-perfect.*

The end of Sect. 3 is devoted to the proof of this result. The case of an irreducible representation is handled in [9, Proposition 4.1]. The proof for an arbitrary tensor product follows the same lines. It is only sketched in *loc. cit.*, and we add quite a few details to Braverman and Gaitsgory's exposition.

### 3.3. Geometric Satake and restriction to a standard Levi subgroup

Consider a subset $J \subseteq I$. In this section we recall Beilinson and Drinfeld's geometric construction of the restriction functor $\mathrm{res}_{M_J}^G$ (see [6, Sect. 5.3]). Additional details can be found in [38, Sect. 8.6] and [4, Sect. 1.15].

Define the root and coroot systems

$$ \Phi_J = \Phi \cap \mathrm{span}_{\mathbb{Z}} \{\alpha_j \mid j \in J\} \quad \text{and} \quad \Phi_J^\vee = \Phi^\vee \cap \mathrm{span}_{\mathbb{Z}} \{\alpha_j^\vee \mid j \in J\} $$

and denote by $\rho_J : \Lambda \to \mathbb{Q}$ the half-sum of the positive coroots in $\Phi_J^\vee$. Then $\rho - \rho_J$ vanishes on $\mathbb{Z}\Phi_J$, so induces a linear form $\rho_{I,J} : \Lambda/\mathbb{Z}\Phi_J \to \mathbb{Q}$.

To $J$ we also attach the standard Levi subgroup $M_J^\vee$ of $G^\vee$. Choose a dominant $\theta_J \in \Lambda$ such that $\langle \alpha_j^\vee, \theta_J \rangle = 0$ for each $j \in J$ and $\langle \alpha_i^\vee, \theta_J \rangle > 0$ for each $i \in I \setminus J$. The embedding

$$ \mathbb{C}^\times \xrightarrow{\theta_J} T^\vee(\mathbb{C}) \to G^\vee(\mathcal{K}) $$

gives rise to an action of $\mathbb{C}^\times$ on $\mathrm{Gr}_J$. Then the set $\mathrm{Gr}_J$ of fixed points under this action can be identified with the affine Grassmannian for $M_J^\vee$. We denote by $\mathrm{Perv}(\mathrm{Gr}_J)$ the category of $M_J^\vee(\mathcal{O})$-equivariant sheaves on $\mathrm{Gr}_J$ supported on finitely many $M_J^\vee(\mathcal{O})$-orbits.

Let $\zeta \in \Lambda/\mathbb{Z}\Phi_J$ be a coset. All the points $L_\mu$ for $\mu \in \zeta$ belong to the same connected component of $\mathrm{Gr}_J$, which we denote by $\mathrm{Gr}_{J,\zeta}$. The map $\zeta \mapsto \mathrm{Gr}_{J,\zeta}$ is a bijection from $\Lambda/\mathbb{Z}\Phi_J$ onto $\pi_0(\mathrm{Gr}_J)$. We denote the stable and unstable sets of $\mathrm{Gr}_{J,\zeta}$ with respect to the $\mathbb{C}^\times$-action by $\mathrm{Gr}_{J,\zeta}^+$ and $\mathrm{Gr}_{J,\zeta}^-$ and form the diagram
In particular, the crystal of an L-perfect basis of an irreducible representation $V(\lambda)$ is unique. We use henceforth the notation $B(\lambda)$ for the associated crystal.

**Remark 3.3.** The crystal $B(\lambda)$ of an irreducible representation $V(\lambda)$ was introduced by Kashiwara in the context of representations of quantum groups. The definition via crystallization at $q = 0$ and the definition via the combinatorics of L-perfect bases yield the same crystal; this follows from [27, Sect. 5].

Fortunately, this nice little theory is not empty. As mentioned in the introduction, any tensor product of irreducible representations has an L-perfect basis, namely its dual canonical basis. Another example of an L-perfect basis: it can be shown, in the case where $G$ is simply laced, that the dual semicanonical basis of an irreducible representation is L-perfect.

**Theorem 3.4.** *The MV basis of a tensor product of irreducible representations is L-perfect.*

The end of Sect. 3 is devoted to the proof of this result. The case of an irreducible representation is handled in [9, Proposition 4.1]. The proof for an arbitrary tensor product follows the same lines. It is only sketched in *loc. cit.*, and we add quite a few details to Braverman and Gaitsgory’s exposition.

### 3.3. Geometric Satake and restriction to a standard Levi subgroup

Consider a subset $J \subseteq I$. In this section we recall Beilinson and Drinfeld’s geometric construction of the restriction functor $\mathrm{res}_{M_J}^G$ (see [6, Sect. 5.3]). Additional details can be found in [38, Sect. 8.6] and [4, Sect. 1.15].

Define the root and coroot systems

$$ \Phi_J = \Phi \cap \mathrm{span}_{\mathbb{Z}} \{\alpha_j \mid j \in J\} \quad \text{and} \quad \Phi_J^\vee = \Phi^\vee \cap \mathrm{span}_{\mathbb{Z}} \{\alpha_j^\vee \mid j \in J\} $$

and denote by $\rho_J : \Lambda \to \mathbb{Q}$ the half-sum of the positive coroots in $\Phi_J^\vee$. Then $\rho - \rho_J$ vanishes on $\mathbb{Z}\Phi_J$, so induces a linear form $\rho_{I,J} : \Lambda/\mathbb{Z}\Phi_J \to \mathbb{Q}$.

To $J$ we also attach the standard Levi subgroup $M_J^\vee$ of $G^\vee$. Choose a dominant $\theta_J \in \Lambda$ such that $\langle \alpha_j^\vee, \theta_J \rangle = 0$ for each $j \in J$ and $\langle \alpha_i^\vee, \theta_J \rangle > 0$ for each $i \in I \setminus J$. The embedding

$$ \mathbb{C}^\times \xrightarrow{\theta_J} T^\vee(\mathbb{C}) \to G^\vee(\mathcal{K}) $$

gives rise to an action of $\mathbb{C}^\times$ on $\mathrm{Gr}_J$. Then the set $\mathrm{Gr}_J$ of fixed points under this action can be identified with the affine Grassmannian for $M_J^\vee$. We denote by $\mathrm{Perv}(\mathrm{Gr}_J)$ the category of $M_J^\vee(\mathcal{O})$-equivariant sheaves on $\mathrm{Gr}_J$ supported on finitely many $M_J^\vee(\mathcal{O})$-orbits.

Let $\zeta \in \Lambda/\mathbb{Z}\Phi_J$ be a coset. All the points $L_\mu$ for $\mu \in \zeta$ belong to the same connected component of $\mathrm{Gr}_J$, which we denote by $\mathrm{Gr}_{J,\zeta}$. The map $\zeta \mapsto \mathrm{Gr}_{J,\zeta}$ is a bijection from $\Lambda/\mathbb{Z}\Phi_J$ onto $\pi_0(\mathrm{Gr}_J)$. We denote the stable and unstable sets of $\mathrm{Gr}_{J,\zeta}$ with respect to the $\mathbb{C}^\times$-action by $\mathrm{Gr}_{J,\zeta}^+$ and $\mathrm{Gr}_{J,\zeta}^-$ and form the diagram
<figure><img src="image_5.png" /></figure>

where $s_{\xi}^J$ and $t_{\xi}^J$ are the inclusion maps and the maps $p_{J,\xi}$ and $q_{J,\xi}$ are defined by

$$p_{J,\xi}(x) = \lim_{c \to 0} \theta_J(c) \cdot x \quad \text{and} \quad q_{J,\xi}(x) = \lim_{c \to \infty} \theta_J(c) \cdot x.$$

Given $\zeta \in \Lambda/\mathbb{Z}\Phi_J$ and $\mathcal{A} \in \text{Perv}(\text{Gr})$, Beilinson and Drinfeld identify the two sheaves

$$(q_{J,\xi})_*(t_{\xi}^J)^!\mathcal{A} \quad \text{and} \quad (p_{J,\xi})_!(s_{\xi}^J)^*\mathcal{A}$$

on $\text{Gr}_{J,\xi}$ via Braden's hyperbolic localization and show that they live in perverse degree $2\rho_{I,J}(\zeta)$. Then they define a functor $r_J^I: \text{Perv}(\text{Gr}) \to \text{Perv}(\text{Gr}_J)$ by

$$r_J^I(\mathcal{A}) = \bigoplus_{\zeta \in \Lambda/\mathbb{Z}\Phi_J} (q_{J,\zeta})_*(t_{\zeta}^J)^! \mathcal{A}[2\rho_{I,J}(\zeta)].$$

For $\mu \in \Lambda$, let $T_{J,\mu}$ be the analog of the unstable subset $T_\mu$ for the affine Grassmannian $\text{Gr}_J$. Let $\zeta$ be the coset of $\mu$ modulo $\mathbb{Z}\Phi_J$ and let $t_{J,\mu}: T_{J,\mu} \to \text{Gr}_{J,\zeta}$ be the inclusion map. Using the Iwasawa decomposition, one checks that $T_\mu \subseteq \text{Gr}_{J,\zeta}^-$ and

$$T_{\mu} = (q_{J,\zeta})^{-1}(T_{J,\mu}). \qquad (5)$$

Performing base change as indicated in the diagram

<figure><img src="image_6.png" /></figure>

we obtain, for any sheaf $\mathcal{A} \in \text{Perv}(\text{Gr})$, a canonical isomorphism

$$H^{2\rho(\mu)}(T_{\mu}, (t_{\mu})^{!}\mathcal{A}) \cong H^{2\rho_J(\mu)}(T_{J,\mu}, (t_{J,\mu})^{!}r_J^I(\mathcal{A})). \qquad (6)$$

For $\mathcal{B} \in \text{Perv}(\text{Gr}_J)$, define

$$F_{J,\mu}(\mathcal{B}) = H^{2\rho_J(\mu)}(T_{J,\mu}, (t_{J,\mu})^!\mathcal{B}) \quad \text{and} \quad F_J(\mathcal{B}) = \bigoplus_{\mu \in \Lambda} F_{J,\mu}(\mathcal{B}).$$

Then (6) can be rewritten as $F_\mu = F_{J,\mu} \circ r_J^I$. This equality can be refined in the following statement: the functor $F_J$ induces an equivalence $\overline{F_J}$ from $\text{Perv}(\text{Gr}_J)$ to the category
<figure><img src="image_4.png" /></figure>

where $s_{\xi}^J$ and $t_{\xi}^J$ are the inclusion maps and the maps $p_{J,\xi}$ and $q_{J,\xi}$ are defined by

$$p_{J,\xi}(x) = \lim_{c \to 0} \theta_J(c) \cdot x \quad \text{and} \quad q_{J,\xi}(x) = \lim_{c \to \infty} \theta_J(c) \cdot x.$$

Given $\zeta \in \Lambda/\mathbb{Z}\Phi_J$ and $\mathcal{A} \in \text{Perv}(\text{Gr})$, Beilinson and Drinfeld identify the two sheaves

$$(q_{J,\xi})_*(t_{\xi}^J)^!\mathcal{A} \quad \text{and} \quad (p_{J,\xi})_!(s_{\xi}^J)^*\mathcal{A}$$

on $\text{Gr}_{J,\xi}$ via Braden's hyperbolic localization and show that they live in perverse degree $2\rho_{I,J}(\zeta)$. Then they define a functor $r_J^I: \text{Perv}(\text{Gr}) \to \text{Perv}(\text{Gr}_J)$ by

$$r_J^I(\mathcal{A}) = \bigoplus_{\zeta \in \Lambda/\mathbb{Z}\Phi_J} (q_{J,\zeta})_*(t_{\zeta}^J)^! \mathcal{A} [2\rho_{I,J}(\zeta)].$$

For $\mu \in \Lambda$, let $T_{J,\mu}$ be the analog of the unstable subset $T_\mu$ for the affine Grassmannian $\text{Gr}_J$. Let $\zeta$ be the coset of $\mu$ modulo $\mathbb{Z}\Phi_J$ and let $t_{J,\mu}: T_{J,\mu} \to \text{Gr}_{J,\zeta}$ be the inclusion map. Using the Iwasawa decomposition, one checks that $T_\mu \subseteq \text{Gr}_{J,\zeta}^-$ and

$$T_{\mu} = (q_{J,\zeta})^{-1}(T_{J,\mu}). \qquad (5)$$

Performing base change as indicated in the diagram

<figure><img src="image_5.png" /></figure>

we obtain, for any sheaf $\mathcal{A} \in \text{Perv}(\text{Gr})$, a canonical isomorphism

$$H^{2\rho(\mu)}(T_{\mu}, (t_{\mu})^{!}\mathcal{A}) \cong H^{2\rho_J(\mu)}(T_{J,\mu}, (t_{J,\mu})^{!}r_J^I(\mathcal{A})). \qquad (6)$$

For $\mathcal{B} \in \text{Perv}(\text{Gr}_J)$, define

$$F_{J,\mu}(\mathcal{B}) = H^{2\rho_J(\mu)}(T_{J,\mu}, (t_{J,\mu})^!\mathcal{B}) \quad \text{and} \quad F_J(\mathcal{B}) = \bigoplus_{\mu \in \Lambda} F_{J,\mu}(\mathcal{B}).$$

Then (6) can be rewritten as $F_\mu = F_{J,\mu} \circ r_J^I$. This equality can be refined in the following statement: the functor $F_J$ induces an equivalence $\overline{F_J}$ from $\text{Perv}(\text{Gr}_J)$ to the category
$\mathrm{Rep}(M_J)$ of finite-dimensional rational representations of $M_J$ and the following diagram commutes:

<figure><img src="image_7.png" /></figure>

### 3.4. The J-decomposition of an MV cycle

We fix a subset $J \subseteq I$. We denote by $P_J^{\neg, \vee}$ the parabolic subgroup of $G^{\vee}$ containing $M_J^{\vee}$ and the negative root subgroups.

The group $P_J^{\neg, \vee}(\mathcal{K})$ certainly acts on $\mathrm{Gr}$; it also acts on $\mathrm{Gr}_J$ via the quotient morphism $P_J^{\neg, \vee}(\mathcal{K}) \to M_J^{\vee}(\mathcal{K})$. Given $\mu \in \Lambda_J^+$, we denote by $\mathrm{Gr}_J^\mu$ the orbit of $L_\mu$ under the action of $M_J^{\vee}(\mathcal{O})$ (or $P_J^{\neg, \vee}(\mathcal{O})$) on $\mathrm{Gr}_J$. Noting that

$$ \lim_{a \to \infty} \theta_J(a) g \theta_J(a)^{-1} = 1 $$

for all $g$ in the unipotent radical of $P_J^{\neg, \vee}$, we see that for any $\zeta \in \Lambda/\mathbb{Z}\Phi_J$, the connected component $\mathrm{Gr}_{J,\zeta}$ of $\mathrm{Gr}_J$ and the unstable subset $\mathrm{Gr}_{J,\zeta}^-$ in $\mathrm{Gr}$ are stable under the action of $P_J^{\neg, \vee}(\mathcal{O})$ and that the map $q_{J,\zeta}$ is equivariant.

Let $\lambda \in (\Lambda^+)^n$, let $\mu \in \Lambda_J^+$ and let $\zeta$ be the coset of $\mu$ modulo $\mathbb{Z}\Phi_J$. We consider the diagram

<figure><img src="image_8.png" /></figure>

The group $G^{\vee}(\mathcal{K})$ acts on $\mathrm{Gr}_n$ by left multiplication on the first factor and the action of the subgroup $G^{\vee}(\mathcal{O})$ leaves $\overline{\mathrm{Gr}}_n^\lambda$ stable. Let $H$ be the stabilizer of $L_\mu$ with respect to the action of $P_J^{\neg, \vee}(\mathcal{O})$ on $\mathrm{Gr}_J$. It acts on $E = \overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(L_\mu)$. Since $q_{J,\zeta} \circ m_n$ is equivariant under the action of $P_J^{\neg, \vee}(\mathcal{O})$, we can make the identification

<figure><img src="image_9.png" /></figure>

where the left vertical arrow is the projection along the first factor. We thereby see that the right vertical arrow is a locally trivial fibration.

In particular, all the fibers $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(x)$ with $x \in \mathrm{Gr}_J^\mu$ are isomorphic varieties. Recalling that $(q_{J,\zeta})^{-1}(L_\mu) \subseteq T_\mu$, we find the following bound for their dimension:

$$ \dim(\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(x)) = \dim E \le \dim(\overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\mu)) = \rho(|\lambda| - \mu). $$
Rep($M_J$) of finite-dimensional rational representations of $M_J$ and the following diagram commutes:

<figure><img src="image_6.png" /></figure>

### 3.4. The J-decomposition of an MV cycle

We fix a subset $J \subseteq I$. We denote by $P_J^{\neg, \vee}$ the parabolic subgroup of $G^{\vee}$ containing $M_J^{\vee}$ and the negative root subgroups.

The group $P_J^{\neg, \vee}(\mathcal{K})$ certainly acts on $Gr_J$; it also acts on $Gr_J$ via the quotient morphism $P_J^{\neg, \vee}(\mathcal{K}) \to M_J^{\vee}(\mathcal{K})$. Given $\mu \in \Lambda_J^+$, we denote by $Gr_J^\mu$ the orbit of $L_\mu$ under the action of $M_J^{\vee}(\mathcal{O})$ (or $P_J^{\neg, \vee}(\mathcal{O})$) on $Gr_J$. Noting that

$$ \lim_{a \to \infty} \theta_J(a) g \theta_J(a)^{-1} = 1 $$

for all $g$ in the unipotent radical of $P_J^{\neg, \vee}$, we see that for any $\zeta \in \Lambda/\mathbb{Z}\Phi_J$, the connected component $Gr_{J,\zeta}$ of $Gr_J$ and the unstable subset $Gr_{J,\zeta}^-$ in $Gr_J$ are stable under the action of $P_J^{\neg, \vee}(\mathcal{O})$ and that the map $q_{J,\zeta}$ is equivariant.

Let $\lambda \in (\Lambda^+)^n$, let $\mu \in \Lambda_J^+$ and let $\zeta$ be the coset of $\mu$ modulo $\mathbb{Z}\Phi_J$. We consider the diagram

<figure><img src="image_7.png" /></figure>

The group $G^{\vee}(\mathcal{K})$ acts on $Gr_n$ by left multiplication on the first factor and the action of the subgroup $G^{\vee}(\mathcal{O})$ leaves $Gr_n^\lambda$ stable. Let $H$ be the stabilizer of $L_\mu$ with respect to the action of $P_J^{\neg, \vee}(\mathcal{O})$ on $Gr_J$. It acts on $E = \overline{Gr_n^\lambda} \cap (q_{J,\zeta} \circ m_n)^{-1}(L_\mu)$. Since $q_{J,\zeta} \circ m_n$ is equivariant under the action of $P_J^{\neg, \vee}(\mathcal{O})$, we can make the identification

<figure><img src="image_8.png" /></figure>

where the left vertical arrow is the projection along the first factor. We thereby see that the right vertical arrow is a locally trivial fibration.

In particular, all the fibers $\overline{Gr}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(x)$ with $x \in \overline{Gr}_J^\mu$ are isomorphic varieties. Recalling that $(q_{J,\zeta})^{-1}(L_\mu) \subseteq T_\mu$, we find the following bound for their dimension:

$$ \dim(\overline{Gr}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(x)) = \dim E \le \dim(\overline{Gr}_n^\lambda \cap (m_n)^{-1}(T_\mu)) = \rho(|\lambda| - \mu). $$
Therefore

$$ \dim(\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (q_{J,\xi} \circ m_{n})^{-1}(\mathrm{Gr}_{J}^{\mu})) \le \dim \mathrm{Gr}_{J}^{\mu} + \rho(|\lambda|-\mu) = 2\rho_{J}(\mu) + \rho(|\lambda|-\mu). \quad (7) $$

Since $\mathrm{Gr}_J^\mu$ is connected and simply-connected, the fibration induces a bijection between the set of irreducible components of $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$ and the set of irreducible components of any fiber $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x)$.

We define

$$ \mathcal{Z}^J(\lambda)_\mu = \{ Z \in \mathrm{Irr}(\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)) \mid \dim Z = 2\rho_J(\mu) + \rho(|\lambda|-\mu) \}. $$

For $\nu \in \Lambda$, we define

$$ \mathcal{Z}_J(\mu)_\nu = \mathrm{Irr}(\overline{\mathrm{Gr}}_J^\mu \cap T_{J,\nu}). $$

As we saw in Sect. 2.2, the map $Z \mapsto Z \cap \mathrm{Gr}_J^\mu$ is a bijection from $\mathcal{Z}_J(\mu)_\nu$ onto the set of irreducible components of $\mathrm{Gr}_J^\mu \cap T_{J,\nu}$.

Fix now $\lambda \in (\Lambda^+)^n$ and $\nu \in \Lambda$. Following Braverman and Gaitsgory, we define a bijection

$$ \mathcal{Z}(\lambda)_\nu \cong \bigsqcup_{\mu \in \Lambda_j^+} \mathcal{Z}^J(\lambda)_\mu \times \mathcal{Z}_J(\mu)_\nu. $$

The union above can be restricted to those weights $\mu$ such that $\mu - \nu \in \mathbb{Z}\Phi_J$, for otherwise $\mathcal{Z}_J(\mu)_\nu$ is empty. Let $\zeta$ denote the coset of $\nu$ modulo $\mathbb{Z}\Phi_J$.

First choose $\mu \in \Lambda_j^+ \cap \zeta$ and a pair $(Z^J, Z_J) \in \mathcal{Z}^J(\lambda)_\mu \times \mathcal{Z}_J(\mu)_\nu$. Using (5) and the fibration above, we see that $Z^J \cap (q_{J,\xi} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu)$ is an irreducible subset of

$$ \overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(T_{J,\nu}) = \overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\nu) $$

of dimension

$$ \dim Z^J - \dim \mathrm{Gr}_J^\mu + \dim Z_J = \rho(|\lambda|-\mu) + \rho_J(\mu-\nu) = \rho(|\lambda|-\nu). $$

Therefore there is a unique $Z \in \mathcal{Z}(\lambda)_\nu$ that contains $Z^J \cap (q_{J,\xi} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu)$ as a dense subset.

Conversely, start from $Z \in \mathcal{Z}(\lambda)_\nu$. Then $Z \subseteq T_\nu \subseteq \overline{\mathrm{Gr}}_{J,\xi}$. We can thus partition Z into locally closed subsets as follows:

$$ Z = \bigsqcup_{\mu \in \Lambda_j^+ \cap \zeta} (Z \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)). $$

Since Z is irreducible, there is a unique $\mu \in \Lambda_j^+ \cap \zeta$ such that $Z \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$ is open dense in Z. That subset is certainly irreducible, hence contained in an irreducible component $Z^J$ of $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$. Also, the composition $q_{J,\xi} \circ m_n$ maps $Z \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$ to an irreducible subset of $\mathrm{Gr}_J^\mu \cap T_{J,\nu}$, which in turn is contained in an irreducible component $Z_J \in \mathcal{Z}_J(\mu)_\nu$. Then

$$ Z \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu) \subseteq Z^J \cap (q_{J,\xi} \circ m_n)^{-1}(Z_J). $$
Therefore

$$ \dim(\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (q_{J,\xi} \circ m_{n})^{-1}(\mathrm{Gr}_{J}^{\mu})) \leq \dim \mathrm{Gr}_{J}^{\mu} + \rho(|\lambda|-\mu) = 2\rho_{J}(\mu) + \rho(|\lambda|-\mu). \quad (7) $$

Since $\mathrm{Gr}_J^\mu$ is connected and simply-connected, the fibration induces a bijection between the set of irreducible components of $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$ and the set of irreducible components of any fiber $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x)$.

We define

$$ \mathcal{Z}^J(\lambda)_\mu = \{ Z \in \mathrm{Irr}(\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)) \mid \dim Z = 2\rho_J(\mu) + \rho(|\lambda|-\mu) \}. $$

For $\nu \in \Lambda$, we define

$$ \mathcal{Z}_J(\mu)_\nu = \mathrm{Irr}(\overline{\mathrm{Gr}}_J^\mu \cap T_{J,\nu}). $$

As we saw in Sect. 2.2, the map $Z \mapsto Z \cap \mathrm{Gr}_J^\mu$ is a bijection from $\mathcal{Z}_J(\mu)_\nu$ onto the set of irreducible components of $\mathrm{Gr}_J^\mu \cap T_{J,\nu}$.

Fix now $\lambda \in (\Lambda^+)^n$ and $\nu \in \Lambda$. Following Braverman and Gaitsgory, we define a bijection

$$ \mathcal{Z}(\lambda)_\nu \cong \bigsqcup_{\mu \in \Lambda_j^+} \mathcal{Z}^J(\lambda)_\mu \times \mathcal{Z}_J(\mu)_\nu. $$

The union above can be restricted to those weights $\mu$ such that $\mu - \nu \in \mathbb{Z}\Phi_J$, for otherwise $\mathcal{Z}_J(\mu)_\nu$ is empty. Let $\zeta$ denote the coset of $\nu$ modulo $\mathbb{Z}\Phi_J$.

First choose $\mu \in \Lambda_j^+ \cap \zeta$ and a pair $(Z^J, Z_J) \in \mathcal{Z}^J(\lambda)_\mu \times \mathcal{Z}_J(\mu)_\nu$. Using (5) and the fibration above, we see that $Z^J \cap (q_{J,\zeta} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu)$ is an irreducible subset of

$$ \overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(T_{J,\nu}) = \overline{\mathrm{Gr}}_n^\lambda \cap (m_n)^{-1}(T_\nu) $$

of dimension

$$ \dim Z^J - \dim \mathrm{Gr}_J^\mu + \dim Z_J = \rho(|\lambda|-\mu) + \rho_J(\mu-\nu) = \rho(|\lambda|-\nu). $$

Therefore there is a unique $Z \in \mathcal{Z}(\lambda)_\nu$ that contains $Z^J \cap (q_{J,\zeta} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu)$ as a dense subset.

Conversely, start from $Z \in \mathcal{Z}(\lambda)_\nu$. Then $Z \subseteq T_\nu \subseteq \overline{\mathrm{Gr}}_{J,\zeta}$. We can thus partition Z into locally closed subsets as follows:

$$ Z = \bigsqcup_{\mu \in \Lambda_j^+ \cap \zeta} (Z \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)). $$

Since Z is irreducible, there is a unique $\mu \in \Lambda_j^+ \cap \zeta$ such that $Z \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$ is open dense in Z. That subset is certainly irreducible, hence contained in an irreducible component $Z^J$ of $\overline{\mathrm{Gr}}_n^\lambda \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$. Also, the composition $q_{J,\zeta} \circ m_n$ maps $Z \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu)$ to an irreducible subset of $\mathrm{Gr}_J^\mu \cap T_{J,\nu}$, which in turn is contained in an irreducible component $Z_J \in \mathcal{Z}_J(\mu)_\nu$. Then

$$ Z \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu) \subseteq Z^J \cap (q_{J,\zeta} \circ m_n)^{-1}(Z_J). $$
The left-hand side has dimension $\rho(|\lambda| - \nu)$ and the right-hand side has dimension

$$ \dim Z^J - 2\rho_J(\mu) + \dim Z_J = \dim Z^J - 2\rho_J(\mu) + \rho(\mu - \nu). $$

Combining this with the bound (7) we get

$$ \dim Z^J = \rho(|\lambda| - \mu) + 2\rho_J(\mu) $$

and therefore $Z^J \in \mathcal{Z}^J(\lambda)_{\mu}$.

These two constructions define mutually inverse bijections; in particular,

$$ Z^J \cap (q_{J,\zeta} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu) = Z \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^\mu). $$

We record that to each MV cycle $Z \in \mathcal{Z}(\lambda)_v$ is assigned a weight $\mu \in \Lambda_J^+$ characterized by the conditions

$$ (q_{J,\zeta} \circ m_n)(Z) \subseteq \overline{\mathrm{Gr}_J^\mu} \quad \text{and} \quad (q_{J,\zeta} \circ m_n)(Z) \cap \mathrm{Gr}_J^\mu \neq \emptyset. $$

This weight will be denoted by $\mu_J(Z)$.

### 3.5. MV bases are L-perfect

We now give the proof of Theorem 3.4, properly speaking. We fix a positive integer $n$ and a tuple $\lambda \in (\Lambda^+)^n$. We need two ingredients besides the constructions explained in Sects. 2.3 and 3.3.

(A) Take $\mathcal{A} \in \mathrm{Perv}(\mathrm{Gr})$ and write the sheaf

$$ \mathcal{B} = r_J^I(\mathcal{A}) = \bigoplus_{\zeta \in \Lambda/\mathbb{Z}\Phi_J} (q_{J,\zeta})_* (t_\zeta^J)^! \mathcal{A}[2\rho_{I,J}(\zeta)] $$

in $\mathrm{Perv}(\mathrm{Gr}_J)$ as a direct sum of isotypic components,

$$ \mathcal{B} = \bigoplus_{\mu \in \Lambda_J^+} \mathrm{IC}(\overline{\mathrm{Gr}_J^\mu}, \mathcal{L}_\mu). \qquad (8) $$

The local systems $\mathcal{L}_\mu$ on $\mathrm{Gr}_J^\mu$ that appear in (8) can be expressed as $\mathcal{L}_\mu = \mathcal{H}^k h^! \mathcal{B}$ where $h: \mathrm{Gr}_J^\mu \to \mathrm{Gr}_J$ is the inclusion map and $k = -\dim \mathrm{Gr}_J^\mu = -2\rho_J(\mu)$. With $e: \{x\} \to \mathrm{Gr}_J^\mu$ the inclusion of a point and $\zeta$ the coset of $\mu$ modulo $\mathbb{Z}\Phi_J$, the fiber of $\mathcal{L}_\mu$ is

$$ (\mathcal{L}_\mu)_x \cong e^! \mathcal{L}_\mu[2 \dim \mathrm{Gr}_J^\mu] \cong H^{2\rho_J(\mu)}(\{x\}, e^! h^! \mathcal{B}). $$

For the specific case

$$ \mathcal{A} = \mathcal{I}_{\lambda} = (m_n)_* \mathrm{IC}(\overline{\mathrm{Gr}_n^\lambda}, \underline{\mathbb{C}}), $$

noting the equality $\rho_J(\mu) + \rho_{I,J}(\zeta) = \rho(\mu)$, we get

$$ (\mathcal{L}_\mu)_x \cong H^{2\rho(\mu)}((q_{J,\zeta} \circ m_n)^{-1}(x), f^! \mathrm{IC}(\overline{\mathrm{Gr}_n^\lambda}, \underline{\mathbb{C}})) $$
The left-hand side has dimension $\rho(|\lambda| - \nu)$ and the right-hand side has dimension

$$ \dim Z^J - 2\rho_J(\mu) + \dim Z_J = \dim Z^J - 2\rho_J(\mu) + \rho(\mu - \nu). $$

Combining this with the bound (7) we get

$$ \dim Z^J = \rho(|\lambda| - \mu) + 2\rho_J(\mu) $$

and therefore $Z^J \in \mathcal{Z}^J(\lambda)_{\mu}$.

These two constructions define mutually inverse bijections; in particular,

$$ Z^J \cap (q_{J,\zeta} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^{\mu}) = Z \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^{\mu}). $$

We record that to each MV cycle $Z \in \mathcal{Z}(\lambda)_v$ is assigned a weight $\mu \in \Lambda_J^+$ characterized by the conditions

$$ (q_{J,\zeta} \circ m_n)(Z) \subseteq \overline{\mathrm{Gr}_J^{\mu}} \quad \text{and} \quad (q_{J,\zeta} \circ m_n)(Z) \cap \mathrm{Gr}_J^{\mu} \neq \emptyset. $$

This weight will be denoted by $\mu_J(Z)$.

### 3.5. MV bases are L-perfect

We now give the proof of Theorem 3.4, properly speaking. We fix a positive integer $n$ and a tuple $\lambda \in (\Lambda^+)^n$. We need two ingredients besides the constructions explained in Sects. 2.3 and 3.3.

(A) Take $\mathcal{A} \in \mathrm{Perv}(\mathrm{Gr})$ and write the sheaf

$$ \mathcal{B} = r_J^I(\mathcal{A}) = \bigoplus_{\zeta \in \Lambda/\mathbb{Z}\Phi_J} (q_{J,\zeta})_* (t_{\zeta}^J)^! \mathcal{A}[2\rho_{I,J}(\zeta)] $$

in $\mathrm{Perv}(\mathrm{Gr}_J)$ as a direct sum of isotypic components,

$$ \mathcal{B} = \bigoplus_{\mu \in \Lambda_J^+} \mathrm{IC}(\overline{\mathrm{Gr}_J^{\mu}}, \mathcal{L}_{\mu}). \qquad (8) $$

The local systems $\mathcal{L}_{\mu}$ on $\mathrm{Gr}_J^{\mu}$ that appear in (8) can be expressed as $\mathcal{L}_{\mu} = \mathcal{H}^k h^! \mathcal{B}$ where $h: \mathrm{Gr}_J^{\mu} \to \mathrm{Gr}_J$ is the inclusion map and $k = -\dim \mathrm{Gr}_J^{\mu} = -2\rho_J(\mu)$. With $e: \{x\} \to \mathrm{Gr}_J^{\mu}$ the inclusion of a point and $\zeta$ the coset of $\mu$ modulo $\mathbb{Z}\Phi_J$, the fiber of $\mathcal{L}_{\mu}$ is

$$ (\mathcal{L}_{\mu})_x \cong e^! \mathcal{L}_{\mu}[2 \dim \mathrm{Gr}_J^{\mu}] \cong H^{2\rho_J(\mu)}(\{x\}, e^! h^! \mathcal{B}). $$

For the specific case

$$ \mathcal{A} = \mathcal{I}_{\lambda} = (m_n)_* \mathrm{IC}(\overline{\mathrm{Gr}_n^{\lambda}}, \underline{\mathbb{C}}), $$

noting the equality $\rho_J(\mu) + \rho_{I,J}(\zeta) = \rho(\mu)$, we get

$$ (\mathcal{L}_{\mu})_x \cong H^{2\rho(\mu)}((q_{J,\zeta} \circ m_n)^{-1}(x), f^! \mathrm{IC}(\overline{\mathrm{Gr}_n^{\lambda}}, \underline{\mathbb{C}})) $$
where $f$ is the injection map depicted in the Cartesian diagram

<figure><img src="image_10.png" /></figure>

The same reasoning as in Sect. 2.3 proves that only the stratum $Gr_n^\lambda$ contributes to this cohomology group. Denoting by $g : \mathrm{Gr}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x) \to \mathrm{Gr}_n^\lambda$ the inclusion map, this observation leads to an isomorphism

$$ (\mathcal{L}_\mu)_x = H^{2\rho_J(\mu)+2\rho_{I,J}(\xi)}(\mathrm{Gr}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x), g! \underline{\mathbb{C}}_{\mathrm{Gr}_n^\lambda} [\dim \mathrm{Gr}_n^\lambda]) \\ \xrightarrow{\cap[\mathrm{Gr}_n^\lambda]} H_{2\rho(|\lambda|-\mu)}^{\mathrm{BM}}(\mathrm{Gr}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x)). $$

Thus the local systems $\mathcal{L}_\mu$ appearing in (8) have a natural basis, namely the set $\mathcal{F}^J(\lambda)_\mu$. We record the following consequence of this discussion: given $(\lambda, \mu, \nu) \in (\Lambda^+)^n \times \Lambda_J^+ \times \Lambda$ such that $\mu - \nu \in \mathbb{Z}\Phi_J$, we have

$$ \begin{aligned} \dim H^{2\rho_J(\nu)}(T_{J,v}, (t_{J,v})^! \mathrm{IC}(\overline{\mathrm{Gr}_J^\mu}, \mathcal{L}_\mu)) &= \mathrm{rank} \mathcal{L}_\mu \times \dim H^{2\rho_J(\nu)}(T_{J,v}, (t_{J,v})^! \mathrm{IC}(\overline{\mathrm{Gr}_J^\mu}, \underline{\mathbb{C}})) \\ &= \mathrm{Card} \mathcal{F}^J(\lambda)_\mu \times \mathrm{Card} \mathcal{F}_J(\mu)_\nu \\ &= \mathrm{Card}\{Z \in \mathcal{F}(\lambda)_\nu \mid \mu_J(Z) = \mu\}. \end{aligned} \quad (9) $$

(B) Now let us start with a sheaf $\mathcal{B}$ in $\mathrm{Perv}(\mathrm{Gr}_J)$ and a weight $\mu \in \Lambda_J^+$. Let us denote by $i : \overline{\mathrm{Gr}_J^\mu} \to \mathrm{Gr}_J$ the inclusion map. By [5, Amplification 1.4.17.1], the largest subobject of $\mathcal{B}$ in $\mathrm{Perv}(\mathrm{Gr}_J)$ supported on $\mathrm{Gr}_J^\mu$ is $\mathcal{B}_{\le J, \mu} = {}^p\tau_{\le 0} i_* i^! \mathcal{B}$, where $^p\tau_{\le 0}$ is the truncation functor for the perverse t-structure. From the distinguished triangle

$$ {}^p\tau_{\le 0} i_* i^! \mathcal{B} \to i_* i^! \mathcal{B} \to {}^p\tau_{>0} i_* i^! \mathcal{B} \xrightarrow{+} $$

in the bounded derived category of constructible sheaves on $\mathrm{Gr}_J$, we deduce the long exact sequence

$$ \begin{aligned} H^{2\rho_J(\nu)-1}(T_{J,v}, (t_{J,v})^{!p} \tau_{>0} i_* i^! \mathcal{B}) &\to H^{2\rho_J(\nu)}(T_{J,v}, (t_{J,v})^{!p} \tau_{\le 0} i_* i^! \mathcal{B}) \\ &\to H^{2\rho_J(\nu)}(T_{J,v}, (t_{J,v})^{!p} i_* i^! \mathcal{B}) \to H^{2\rho_J(\nu)}(T_{J,v}, (t_{J,v})^{!p} \tau_{>0} i_* i^! \mathcal{B}). \end{aligned} $$

Theorem 3.5 in [39] implies that the two extreme terms vanish, and therefore

$$ F_{J,v}(\mathcal{B}_{\le J, \mu}) = H^{2\rho_J(\nu)}(T_{J,v}, (t_{J,v})^{!p} i_* i^! \mathcal{B}). $$
where $f$ is the injection map depicted in the Cartesian diagram

<figure><img src="image_9.png" /></figure>

The same reasoning as in Sect. 2.3 proves that only the stratum $\mathrm{Gr}_n^\lambda$ contributes to this cohomology group. Denoting by $g : \mathrm{Gr}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x) \to \mathrm{Gr}_n^\lambda$ the inclusion map, this observation leads to an isomorphism

$$ (\mathcal{L}_\mu)_x = H^{2\rho_J(\mu)+2\rho_{I,J}(\xi)}(\mathrm{Gr}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x), g! \underline{\mathbb{C}}_{\mathrm{Gr}_n^\lambda} [\dim \mathrm{Gr}_n^\lambda]) \\ \xrightarrow{\cap[\mathrm{Gr}_n^\lambda]} H_{2\rho(|\lambda|-\mu)}^{\mathrm{BM}}(\mathrm{Gr}_n^\lambda \cap (q_{J,\xi} \circ m_n)^{-1}(x)). $$

Thus the local systems $\mathcal{L}_\mu$ appearing in (8) have a natural basis, namely the set $\mathcal{F}^J(\lambda)_\mu$. We record the following consequence of this discussion: given $(\lambda, \mu, \nu) \in (\Lambda^+)^n \times \Lambda_J^+ \times \Lambda$ such that $\mu - \nu \in \mathbb{Z}\Phi_J$, we have

$$ \begin{aligned} \dim H^{2\rho_J(\nu)}(T_{J,\nu}, (t_{J,\nu})^! \mathrm{IC}(\overline{\mathrm{Gr}_J^\mu}, \mathcal{L}_\mu)) &= \mathrm{rank} \mathcal{L}_\mu \times \dim H^{2\rho_J(\nu)}(T_{J,\nu}, (t_{J,\nu})^! \mathrm{IC}(\overline{\mathrm{Gr}_J^\mu}, \underline{\mathbb{C}})) \\ &= \mathrm{Card} \mathcal{F}^J(\lambda)_\mu \times \mathrm{Card} \mathcal{F}_J(\mu)_\nu \\ &= \mathrm{Card}\{Z \in \mathcal{F}(\lambda)_\nu \mid \mu_J(Z) = \mu\}. \end{aligned} \quad (9) $$

(B) Now let us start with a sheaf $\mathcal{B}$ in $\mathrm{Perv}(\mathrm{Gr}_J)$ and a weight $\mu \in \Lambda_J^+$. Let us denote by $i : \overline{\mathrm{Gr}_J^\mu} \to \mathrm{Gr}_J$ the inclusion map. By [5, Amplification 1.4.17.1], the largest subobject of $\mathcal{B}$ in $\mathrm{Perv}(\mathrm{Gr}_J)$ supported on $\mathrm{Gr}_J^\mu$ is $\mathcal{B}_{\le J, \mu} = {}^p\tau_{\le 0} i_* i^! \mathcal{B}$, where ${}^p\tau_{\le 0}$ is the truncation functor for the perverse $t$-structure. From the distinguished triangle

$$ {}^p\tau_{\le 0} i_* i^! \mathcal{B} \to i_* i^! \mathcal{B} \xrightarrow{+} {}^p\tau_{> 0} i_* i^! \mathcal{B} $$

in the bounded derived category of constructible sheaves on $\mathrm{Gr}_J$, we deduce the long exact sequence

$$ \begin{aligned} H^{2\rho_J(\nu)-1}(T_{J,\nu}, (t_{J,\nu})^{!p} \tau_{>0} i_* i^! \mathcal{B}) &\to H^{2\rho_J(\nu)}(T_{J,\nu}, (t_{J,\nu})^{!p} \tau_{\le 0} i_* i^! \mathcal{B}) \\ &\to H^{2\rho_J(\nu)}(T_{J,\nu}, (t_{J,\nu})^{!p} i_* i^! \mathcal{B}) \to H^{2\rho_J(\nu)}(T_{J,\nu}, (t_{J,\nu})^{!p} \tau_{>0} i_* i^! \mathcal{B}). \end{aligned} $$

Theorem 3.5 in [39] implies that the two extreme terms vanish, and therefore

$$ F_{J,\nu}(\mathcal{B}_{\le J, \mu}) = H^{2\rho_J(\nu)}(T_{J,\nu}, (t_{J,\nu})^{!p} i_* i^! \mathcal{B}). $$
Let us patch all these pieces together. We take $(\lambda, \mu, \nu) \in (\Lambda^+)^n \times \Lambda_J^+ \times \Lambda$ such that $\mu$ and $\nu$ belong to the same coset $\zeta \in Z/\mathbb{Z}\Phi_J$ and we consider

$$ \mathcal{I}_{\lambda} = (m_n)_* \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \underline{\mathcal{C}}) \quad \text{and} \quad \mathcal{B} = r_J^I(\mathcal{I}_{\lambda}). $$

Composing the isomorphisms given in (6) and (3), we get

$$ \begin{aligned} H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^! \mathcal{B}) &\cong H^{2\rho(v)}(T_v, (t_v)^! \mathcal{I}_{\lambda}) \\ &\cong H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (m_n)^{-1}(T_v)). \end{aligned} \quad (10) $$

To save space we set $S = (q_{J,\zeta})^{-1}(\overline{\mathrm{Gr}}_{J,\zeta}^{\mu}) \cap T_v$ and denote by $s: S \to T_v$ the inclusion map. Chasing in the three-dimensional figure

<figure><img src="image_11.png" /></figure>

we complete (10) to the following commutative diagram:

$$ \begin{aligned} H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^! i_* t^! \mathcal{B}) &\xrightarrow{\sim} H^{2\rho(v)}(S, (t_v S)^! \mathcal{I}_{\lambda}) \xrightarrow{\sim} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (m_n)^{-1}(S)) \\ H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^! \mathcal{B}) &\xrightarrow{\sim} H^{2\rho(v)}(T_v, (t_v)^! \mathcal{I}_{\lambda}) \xrightarrow{\sim} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (m_n)^{-1}(T_v)) \end{aligned} $$

As explained in (B), the left vertical arrow of this diagram is the inclusion map

$$ F_{J,v}(\mathcal{B}_{\le J, \mu}) \to F_{J,v}(\mathcal{B}). $$

If an MV cycle $Z \in \mathcal{Z}(\lambda)_v$ satisfies

$$ \mu_J(Z) \le J \mu, $$

then it is contained in $(m_n)^{-1}(S)$, so the fundamental class of $Z \cap \mathrm{Gr}_n^\lambda$ belongs to $F_{J,v}(\mathcal{B}_{\le J, \mu})$. Looking at equation (9), we see that there are just enough such MV cycles to span this subspace. Going through the geometric Satake correspondence, we conclude that the MV basis of $\mathcal{V}(\lambda)$ satisfies condition (P1) for being L-perfect.
Let us patch all these pieces together. We take $(\lambda, \mu, \nu) \in (\Lambda^+)^n \times \Lambda_J^+ \times \Lambda$ such that $\mu$ and $\nu$ belong to the same coset $\zeta \in Z/\mathbb{Z}\Phi_J$ and we consider

$$ \mathcal{I}_{\lambda} = (m_n)_* \mathrm{IC}(\overline{\mathrm{Gr}}_{n}^{\lambda}, \underline{\mathcal{C}}) \quad \text{and} \quad \mathcal{B} = r_J^I(\mathcal{I}_{\lambda}). $$

Composing the isomorphisms given in (6) and (3), we get

$$ \begin{aligned} H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^! \mathcal{B}) &\cong H^{2\rho(v)}(T_v, (t_v)^! \mathcal{I}_{\lambda}) \\ &\cong H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (m_n)^{-1}(T_v)). \end{aligned} \quad (10) $$

To save space we set $S = (q_{J,\zeta})^{-1}(\overline{\mathrm{Gr}}_J^\mu) \cap T_v$ and denote by $s: S \to T_v$ the inclusion map. Chasing in the three-dimensional figure

<figure><img src="image_10.png" /></figure>

we complete (10) to the following commutative diagram:

$$ \begin{aligned} H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^! i_* t^! \mathcal{B}) &\xrightarrow{\sim} H^{2\rho(v)}(S, (t_v S)^! \mathcal{I}_{\lambda}) \xrightarrow{\sim} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (m_n)^{-1}(S)) \\ H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^! \mathcal{B}) &\xrightarrow{\sim} H^{2\rho(v)}(T_v, (t_v)^! \mathcal{I}_{\lambda}) \xrightarrow{\sim} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (m_n)^{-1}(T_v)) \end{aligned} $$

As explained in (B), the left vertical arrow of this diagram is the inclusion map

$$ F_{J,v}(\mathcal{B}_{\le J} \mu) \to F_{J,v}(\mathcal{B}). $$

If an MV cycle $Z \in \mathcal{Z}(\lambda)_v$ satisfies

$$ \mu_J(Z) \le_J \mu, $$

then it is contained in $(m_n)^{-1}(S)$, so the fundamental class of $Z \cap \mathrm{Gr}_n^\lambda$ belongs to $F_{J,v}(\mathcal{B}_{\le J} \mu)$. Looking at equation (9), we see that there are just enough such MV cycles to span this subspace. Going through the geometric Satake correspondence, we conclude that the MV basis of $\mathcal{V}(\lambda)$ satisfies condition (P1) for being L-perfect.
Turning now to condition (P2), we consider the diagram below, consisting of inclusion maps:

<figure><img src="image_12.png" /></figure>

For any sheaf $\mathcal{F} \in \mathrm{Perv}(\mathrm{Gr}_J)$ supported on $\overline{\mathrm{Gr}}_J^{\mu} \setminus \mathrm{Gr}_J^{\mu}$, we have

$$ \mathrm{Hom}(\mathcal{F}, {}^p\tau_{\le 0}(h_*h^!\mathcal{B})) = \mathrm{Hom}(\mathcal{F}, h_*h^!\mathcal{B}) = \mathrm{Hom}(h^*\mathcal{F}, h^!\mathcal{B}) = 0 $$

in the bounded derived category of constructible sheaves over $\mathrm{Gr}_J$ (the first two equalities by adjunction, the last one because $h^*\mathcal{F} = 0$). Since $h_*h^!\mathcal{B}$ is concentrated in nonnegative perverse degrees (see [5, Proposition 1.4.16]), the sheaf ${}^p\tau_{\le 0}(h_*h^!\mathcal{B})$ is perverse, and from the semisimplicity of $\mathrm{Perv}(\mathrm{Gr}_J)$ we conclude that

$$ \mathrm{Hom}({}^p\tau_{\le 0}(h_*h^!\mathcal{B}), \mathcal{F}) = 0. $$

Again, in the distinguished triangle

$$ i_*f_*f^!i^!\mathcal{B} \to i_*i^!\mathcal{B} \to i_*g_*g^!i^!\mathcal{B} \xrightarrow{+} $$

all sheaves are concentrated in nonnegative perverse degrees. Denoting by ${}^pH^1$ the first homology group for the perverse t-structure, we obtain the exact sequence

$$ 0 \to {}^p\tau_{\le 0}(i_*f_*f^!i^!\mathcal{B}) \to {}^p\tau_{\le 0}(i_*i^!\mathcal{B}) \to {}^p\tau_{\le 0}(h_*h^!\mathcal{B}) \to {}^pH^1(i_*f_*f^!i^!\mathcal{B}). $$

The perverse sheaf on the right is supported on $\overline{\mathrm{Gr}}_J^{\mu} \setminus \mathrm{Gr}_J^{\mu}$, so the right arrow is zero by the previous step. The resulting short exact sequence can be identified with

$$ 0 \to \mathcal{B}_{<J\mu} \to \mathcal{B}_{\le J\mu} \to \mathcal{B}_{\le J\mu}/\mathcal{B}_{<J\mu} \to 0. $$

With the same arguments as in point (B) above, we deduce that

$$ F_{J,v}(\mathcal{B}_{\le J\mu}/\mathcal{B}_{<J\mu}) = F_{J,v}({}^p\tau_{\le 0}h_*h^!\mathcal{B}) = H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^!h_*h^!\mathcal{B}). $$

In (11), we replace $\overline{\mathrm{Gr}}_J^{\mu}$ by $\mathrm{Gr}_J^{\mu}$; the same chasing as before now leads to the isomorphism

$$ H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^!h_*h^!\mathcal{B}) \xrightarrow{\cong} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^{\mu}) \cap (m_n)^{-1}(T_v)). \quad (12) $$

Now point (A) at the beginning of this section explains that $\mathcal{H}^k h^! \mathcal{B}$, where $k = -2\rho_J(\mu)$, is the local system $\mathcal{L}_\mu$ and it comes with a natural basis, namely $\mathcal{L}^J(\lambda)_\mu$. This basis induces a decomposition of $\mathcal{B}_{\le J\mu}/\mathcal{B}_{<J\mu}$ into a sum of simple objects in $\mathrm{Perv}(\mathrm{Gr}_J)$. On
Turning now to condition (P2), we consider the diagram below, consisting of inclusion maps:

<figure><img src="image_11.png" /></figure>

For any sheaf $\mathcal{F} \in \mathrm{Perv}(\mathrm{Gr}_J)$ supported on $\overline{\mathrm{Gr}}_J^{\mu} \setminus \mathrm{Gr}_J^{\mu}$, we have

$$ \mathrm{Hom}(\mathcal{F}, {}^p\tau_{\le 0}(h_*h^!\mathcal{B})) = \mathrm{Hom}(\mathcal{F}, h_*h^!\mathcal{B}) = \mathrm{Hom}(h^*\mathcal{F}, h^!\mathcal{B}) = 0 $$

in the bounded derived category of constructible sheaves over $\mathrm{Gr}_J$ (the first two equalities by adjunction, the last one because $h^*\mathcal{F} = 0$). Since $h_*h^!\mathcal{B}$ is concentrated in nonnegative perverse degrees (see [5, Proposition 1.4.16]), the sheaf ${}^p\tau_{\le 0}(h_*h^!\mathcal{B})$ is perverse, and from the semisimplicity of $\mathrm{Perv}(\mathrm{Gr}_J)$ we conclude that

$$ \mathrm{Hom}({}^p\tau_{\le 0}(h_*h^!\mathcal{B}), \mathcal{F}) = 0. $$

Again, in the distinguished triangle

$$ i_*f_*f^!i^!\mathcal{B} \to i_*i^!\mathcal{B} \to i_*g_*g^!i^!\mathcal{B} \xrightarrow{+} $$

all sheaves are concentrated in nonnegative perverse degrees. Denoting by ${}^pH^1$ the first homology group for the perverse t-structure, we obtain the exact sequence

$$ 0 \to {}^p\tau_{\le 0}(i_*f_*f^!i^!\mathcal{B}) \to {}^p\tau_{\le 0}(i_*i^!\mathcal{B}) \to {}^p\tau_{\le 0}(h_*h^!\mathcal{B}) \to {}^pH^1(i_*f_*f^!i^!\mathcal{B}). $$

The perverse sheaf on the right is supported on $\overline{\mathrm{Gr}}_J^{\mu} \setminus \mathrm{Gr}_J^{\mu}$, so the right arrow is zero by the previous step. The resulting short exact sequence can be identified with

$$ 0 \to \mathcal{B}_{<J\mu} \to \mathcal{B}_{\le J\mu} \to \mathcal{B}_{\le J\mu}/\mathcal{B}_{<J\mu} \to 0. $$

With the same arguments as in point (B) above, we deduce that

$$ F_{J,v}(\mathcal{B}_{\le J\mu}/\mathcal{B}_{<J\mu}) = F_{J,v}({}^p\tau_{\le 0}(h_*h^!\mathcal{B})) = H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^!h_*h^!\mathcal{B}). $$

In (11), we replace $\overline{\mathrm{Gr}}_J^{\mu}$ by $\mathrm{Gr}_J^{\mu}$; the same chasing as before now leads to the isomorphism

$$ H^{2\rho_J(v)}(T_{J,v}, (t_{J,v})^!h_*h^!\mathcal{B}) \xrightarrow{\cong} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap (q_{J,\zeta} \circ m_n)^{-1}(\mathrm{Gr}_J^{\mu}) \cap (m_n)^{-1}(T_v)). \quad (12) $$

Now point (A) at the beginning of this section explains that $\mathcal{H}^k h^! \mathcal{B}$, where $k = -2\rho_J(\mu)$, is the local system $\mathcal{L}_\mu$ and it comes with a natural basis, namely $\mathcal{L}^J(\lambda)_\mu$. This basis induces a decomposition of $\mathcal{B}_{\le J\mu}/\mathcal{B}_{<J\mu}$ into a sum of simple objects in $\mathrm{Perv}(\mathrm{Gr}_J)$. On
the one hand, this decomposition can be followed through the geometric Satake correspondence, where it gives a decomposition of the subquotient of the isotypic filtration of $\mathrm{res}_{M_J}^G V(\lambda)$ into a direct sum of irreducible representations. On the other hand, it can also be tracked through the isomorphism (12):

$$F_{J,v}(\mathcal{B}_{\le J, \mu} / \mathcal{B}_{< J, \mu}) \cong \bigoplus_{Y \in \mathcal{Y}^J(\lambda)_{\mu}} H_{2\rho(|\lambda|-v)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap Y \cap (m_n)^{-1}(T_v)). \quad (13)$$

From Sect. 3.4, we see that the irreducible components of

$$\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (q_{J, \xi} \circ m_{n})^{-1}(\mathrm{Gr}_{J}^{\mu}) \cap (m_{n})^{-1}(T_{\nu})$$

of dimension $\rho(|\lambda| - v)$ are the cycles $Z^J \cap (q_{J,\xi} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu)$ with $(Z^J, Z_J) \in \mathcal{Z}^J(\lambda)_\mu \times \mathcal{Z}_J(\mu)_v$. The basis of the right-hand side of (12) afforded by the fundamental classes of these irreducible components is thus compatible with the decomposition (13). Therefore, the MV basis of $V(\lambda)$ satisfies condition (P2) for being L-perfect.

The proof of Theorem 3.4 is now complete.

**Remark 3.5.** The proof establishes that the MV basis of $V(\lambda)$ satisfies a stronger property than (P2): there exists an isomorphism of the isotypic component $V(\lambda)_{J,\mu}$ with a direct sum of copies of the irreducible representation $V_J(\mu)$ such that the induced basis on $V(\lambda)_{J,\mu}$ matches the direct sum of the MV bases of the summands.

### 3.6. Crystal structure on MV cycles

Let $\lambda \in (\Lambda^+)^n$. The MV basis of $V(\lambda)$ defined in Sect. 2.3 is indexed by

$$\mathcal{Z}(\lambda) = \bigsqcup_{v \in \Lambda} \mathcal{Z}(\lambda)_v$$

and is L-perfect. Thus, the set $\mathcal{Z}(\lambda)$ is endowed with the structure of a crystal, as explained in Sect. 3.2. Obviously the weight of an MV cycle $Z \in \mathcal{Z}(\lambda)_v$ is simply $\mathrm{wt}(Z) = v$. The aim of this section is to characterize the action on $\mathcal{Z}(\lambda)$ of the operators $\tilde{e}_i$ and $\tilde{f}_i$.

In semisimple rank 1, one can give an explicit analytical description of the MV cycles, as follows.

**Proposition 3.6.** Assume that $G$ has semisimple rank 1 and denote by $\alpha$ and $\alpha^\vee$ the positive root and coroot. Let $y : \mathbb{G}_a \to G^\vee$ be the additive one-parameter subgroup for the root $-\alpha^\vee$. Let $(\mu, v) \in \Lambda^+ \times \Lambda$ and set $r = \langle \alpha^\vee, \mu \rangle$. Then $\overline{\mathrm{Gr}}^\mu \cap T_v$ is nonempty if and only if there exists $p \in \{0, 1, \dots, r\}$ such that $v = \mu - p\alpha$; in this case, the map $a \mapsto y(az^{p-r})L_v$ induces an isomorphism of algebraic varieties

$$\mathcal{O}/z^p\mathcal{O} \xrightarrow{\sim} \overline{\mathrm{Gr}}^\mu \cap T_v,$$

so $\overline{\mathrm{Gr}}^\mu \cap T_v$ is an affine space of dimension $p$ and $\mathcal{Z}(\mu)_v$ is a singleton.
the one hand, this decomposition can be followed through the geometric Satake correspondence, where it gives a decomposition of the subquotient of the isotypic filtration of $\mathrm{res}_{M_J}^G V(\lambda)$ into a direct sum of irreducible representations. On the other hand, it can also be tracked through the isomorphism (12):

$$F_{J,\nu}(\mathcal{B}_{\le J, \mu} / \mathcal{B}_{< J, \mu}) \cong \bigoplus_{Y \in \mathcal{Y}^J(\lambda)_{\mu}} H_{2\rho(|\lambda|-\nu)}^{\mathrm{BM}}(\mathrm{Gr}_n^{\lambda} \cap Y \cap (m_n)^{-1}(T_{\nu})). \quad (13)$$

From Sect. 3.4, we see that the irreducible components of

$$\overline{\mathrm{Gr}}_{n}^{\lambda} \cap (q_{J, \zeta} \circ m_{n})^{-1}(\mathrm{Gr}_{J}^{\mu}) \cap (m_{n})^{-1}(T_{\nu})$$

of dimension $\rho(|\lambda| - \nu)$ are the cycles $Z^J \cap (q_{J,\zeta} \circ m_n)^{-1}(Z_J \cap \mathrm{Gr}_J^\mu)$ with $(Z^J, Z_J) \in \mathcal{Y}^J(\lambda)_\mu \times \mathcal{Y}_J(\mu)_\nu$. The basis of the right-hand side of (12) afforded by the fundamental classes of these irreducible components is thus compatible with the decomposition (13). Therefore, the MV basis of $V(\lambda)$ satisfies condition (P2) for being L-perfect.

The proof of Theorem 3.4 is now complete.

**Remark 3.5.** The proof establishes that the MV basis of $V(\lambda)$ satisfies a stronger property than (P2): there exists an isomorphism of the isotypic component $V(\lambda)_{J,\mu}$ with a direct sum of copies of the irreducible representation $V_J(\mu)$ such that the induced basis on $V(\lambda)_{J,\mu}$ matches the direct sum of the MV bases of the summands.

### 3.6. Crystal structure on MV cycles

Let $\lambda \in (\Lambda^+)^n$. The MV basis of $V(\lambda)$ defined in Sect. 2.3 is indexed by

$$\mathcal{L}(\lambda) = \bigsqcup_{\nu \in \Lambda} \mathcal{L}(\lambda)_\nu$$

and is L-perfect. Thus, the set $\mathcal{L}(\lambda)$ is endowed with the structure of a crystal, as explained in Sect. 3.2. Obviously the weight of an MV cycle $Z \in \mathcal{L}(\lambda)_\nu$ is simply $\mathrm{wt}(Z) = \nu$. The aim of this section is to characterize the action on $\mathcal{L}(\lambda)$ of the operators $\tilde{e}_i$ and $\tilde{f}_i$.

In semisimple rank 1, one can give an explicit analytical description of the MV cycles, as follows.

**Proposition 3.6.** Assume that $G$ has semisimple rank 1 and denote by $\alpha$ and $\alpha^\vee$ the positive root and coroot. Let $y : \mathbb{G}_a \to G^\vee$ be the additive one-parameter subgroup for the root $-\alpha^\vee$. Let $(\mu, \nu) \in \Lambda^+ \times \Lambda$ and set $r = \langle \alpha^\vee, \mu \rangle$. Then $\overline{\mathrm{Gr}}^\mu \cap T_\nu$ is nonempty if and only if there exists $p \in \{0, 1, \dots, r\}$ such that $\nu = \mu - p\alpha$; in this case, the map $a \mapsto y(az^{p-r})L_\nu$ induces an isomorphism of algebraic varieties

$$\mathcal{O}/z^p\mathcal{O} \xrightarrow{\sim} \overline{\mathrm{Gr}}^\mu \cap T_\nu,$$

so $\overline{\mathrm{Gr}}^\mu \cap T_\nu$ is an affine space of dimension $p$ and $\mathcal{L}(\mu)_\nu$ is a singleton.
We skip the proof since this proposition is well-known; compare for instance with [2, Proposition 3.10]. We can now describe the crystal structure on $\mathcal{L}(\lambda)$, which extends [2, Proposition 4.2].

**Proposition 3.7.** Let $(\lambda, v) \in (\Lambda^+)^n \times \Lambda$, let $i \in I$ and let $Z \in \mathcal{L}(\lambda)_v$.

(i) We have $\mathrm{wt}(Z) = v$, $\varepsilon_i(Z) = \frac{1}{2}\langle \alpha_i^\vee, \mu_{\{i\}}(Z) - v \rangle$ and $\varphi_i(Z) = \frac{1}{2}\langle \alpha_i^\vee, \mu_{\{i\}}(Z) + v \rangle$.

(ii) Let $Y \in \mathcal{L}(\lambda)_{v+\alpha_i}$. Then $Y = \tilde{e}_i Z$ if and only if $Y \subseteq \bar{Z}$ and $\mu_{\{i\}}(Y) = \mu_{\{i\}}(Z)$.

*Proof.* Let $\lambda, v, i, Z$ as in the statement and set $\mu = \mu_{\{i\}}(Z)$. By definition, the MV cycles $\tilde{e}_i Z$ and $\tilde{f}_i Z$ (if nonzero) are obtained by letting the Chevalley generators $e_i$ and $f_i$ act on (the basis element indexed by) $Z$ in the appropriate subquotient of the isotypic filtration of $\mathrm{res}_{M_{\{i\}}}^G V(\lambda)$. According to (13), this entails that

$$ \mu_{\{i\}}(Z) = \mu_{\{i\}}(\tilde{e}_i Z) = \mu_{\{i\}}(\tilde{f}_i Z) \quad \text{and} \quad Z^{\{i\}} = (\tilde{e}_i Z)^{\{i\}} = (\tilde{f}_i Z)^{\{i\}}. $$

In addition, $\mathcal{L}_{\{i\}}(\mu)_{v+\alpha_i}$ and $\mathcal{L}_{\{i\}}(\mu)_{v-\alpha_i}$ are empty or singletons, and the MV cycles $(\tilde{e}_i Z)_{\{i\}}$ and $(\tilde{f}_i Z)_{\{i\}}$ in the affine Grassmannian $\mathrm{Gr}_{\{i\}}$ are uniquely determined by weight considerations.

The statements can be deduced from these facts by using the explicit description provided by Proposition 3.6 and the construction of the map $(Z^J, Z_J) \mapsto Z$ in Sect. 3.4. ■

# 4. The path model and MV cycles

In the previous section, we defined the structure of a crystal on the set $\mathcal{L}(\lambda)$. In this section, we turn to Littelmann's path model [33] to study this structure. This combinatorial device can be used to effectively assemble MV cycles. Our construction is inspired by the results presented in [17] but is more flexible, for it relaxes the restriction to minimal galleries.

In this paper, piecewise linear means continuous piecewise linear. We keep the notation set up at the opening of Sect. 3.

## 4.1. Recollections on the path model

Let $\Lambda_R = \Lambda \otimes_Z \mathbb{R}$ be the real vector space spanned by the weight lattice and let $\Lambda_R^+$ be the dominant cone inside $\Lambda_R$.

A *path* is a piecewise linear map $\pi : [0, 1] \to \Lambda_R$ such that $\pi(0) = 0$ and $\pi(1) \in \Lambda$. We denote by $\tilde{\Pi}$ the set of all paths. The concatenation $\pi * \eta$ of two paths is defined in the usual way: $\pi * \eta(t) = \pi(2t)$ for $0 \le t \le 1/2$, and $\pi * \eta(t) = \eta(2t - 1) + \pi(1)$ for $1/2 \le t \le 1$.

In [33], the third author associates to each simple root $\alpha$ of $\Phi$ a pair $(e_\alpha, f_\alpha)$ of “root operators” from $\tilde{\Pi}$ to $\tilde{\Pi} \cup \{0\}$ and shows that the construction yields a seminormal crystal
We skip the proof since this proposition is well-known; compare for instance with [2, Proposition 3.10]. We can now describe the crystal structure on $\mathcal{L}(\lambda)$, which extends [2, Proposition 4.2].

**Proposition 3.7.** Let $(\lambda, v) \in (\Lambda^+)^n \times \Lambda$, let $i \in I$ and let $Z \in \mathcal{L}(\lambda)_v$.

(i) We have $\mathrm{wt}(Z) = v$, $\varepsilon_i(Z) = \frac{1}{2}\langle \alpha_i^\vee, \mu_{\{i\}}(Z) - v \rangle$ and $\varphi_i(Z) = \frac{1}{2}\langle \alpha_i^\vee, \mu_{\{i\}}(Z) + v \rangle$.

(ii) Let $Y \in \mathcal{L}(\lambda)_{v+\alpha_i}$. Then $Y = \tilde{e}_i Z$ if and only if $Y \subseteq \bar{Z}$ and $\mu_{\{i\}}(Y) = \mu_{\{i\}}(Z)$.

*Proof.* Let $\lambda, v, i, Z$ as in the statement and set $\mu = \mu_{\{i\}}(Z)$. By definition, the MV cycles $\tilde{e}_i Z$ and $\tilde{f}_i Z$ (if nonzero) are obtained by letting the Chevalley generators $e_i$ and $f_i$ act on (the basis element indexed by) $Z$ in the appropriate subquotient of the isotypic filtration of $\mathrm{res}_{M_{\{i\}}}^G V(\lambda)$. According to (13), this entails that

$$ \mu_{\{i\}}(Z) = \mu_{\{i\}}(\tilde{e}_i Z) = \mu_{\{i\}}(\tilde{f}_i Z) \quad \text{and} \quad Z^{\{i\}} = (\tilde{e}_i Z)^{\{i\}} = (\tilde{f}_i Z)^{\{i\}}. $$

In addition, $\mathcal{L}_{\{i\}}(\mu)_{v+\alpha_i}$ and $\mathcal{L}_{\{i\}}(\mu)_{v-\alpha_i}$ are empty or singletons, and the MV cycles $(\tilde{e}_i Z)_{\{i\}}$ and $(\tilde{f}_i Z)_{\{i\}}$ in the affine Grassmannian $\mathrm{Gr}_{\{i\}}$ are uniquely determined by weight considerations.

The statements can be deduced from these facts by using the explicit description provided by Proposition 3.6 and the construction of the map $(Z^J, Z_J) \mapsto Z$ in Sect. 3.4. ■

# 4. The path model and MV cycles

In the previous section, we defined the structure of a crystal on the set $\mathcal{L}(\lambda)$. In this section, we turn to Littelmann's path model [33] to study this structure. This combinatorial device can be used to effectively assemble MV cycles. Our construction is inspired by the results presented in [17] but is more flexible, for it relaxes the restriction to minimal galleries.

In this paper, piecewise linear means continuous piecewise linear. We keep the notation set up at the opening of Sect. 3.

## 4.1. Recollections on the path model

Let $\Lambda_R = \Lambda \otimes_Z \mathbb{R}$ be the real vector space spanned by the weight lattice and let $\Lambda_R^+$ be the dominant cone inside $\Lambda_R$.

A *path* is a piecewise linear map $\pi : [0, 1] \to \Lambda_R$ such that $\pi(0) = 0$ and $\pi(1) \in \Lambda$. We denote by $\tilde{\Pi}$ the set of all paths. The concatenation $\pi * \eta$ of two paths is defined in the usual way: $\pi * \eta(t) = \pi(2t)$ for $0 \le t \le 1/2$, and $\pi * \eta(t) = \eta(2t - 1) + \pi(1)$ for $1/2 \le t \le 1$.

In [33], the third author associates to each simple root $\alpha$ of $\Phi$ a pair $(e_\alpha, f_\alpha)$ of “root operators” from $\tilde{\Pi}$ to $\tilde{\Pi} \cup \{0\}$ and shows that the construction yields a seminormal crystal
structure on $\tilde{\Pi}$. Here the weight map is given by $wt(\pi) = \pi(1)$. To agree with the notation in Sect. 3.1, we will write $\tilde{e}_i$ and $\tilde{f}_i$ instead of $e_{\alpha_i}$ and $f_{\alpha_i}$ for each $i \in I$.

Let $\ell: [0, 1] \to \mathbb{R}$ be a piecewise linear function. We say that $p \in \mathbb{R}$ is a local absolute minimum of $\ell$ if there exists a compact interval $[a, b] \subseteq [0, 1]$ over which $\ell$ takes the value $p$, and there exists an $\epsilon > 0$ such that $\ell(x) > p$ for all $x \in (a - \epsilon, a) \cap [0, 1]$ and all $x \in (b, b + \epsilon) \cap [0, 1]$.

Given $\pi \in \tilde{\Pi}$, we denote by $A\pi$ the set of all paths $\eta \in \tilde{\Pi}$ that can be obtained from $\pi$ by applying a finite sequence of root operators $\tilde{e}_i$ or $\tilde{f}_i$. We say that a path $\pi \in \tilde{\Pi}$ is integral if for each $\eta \in A\pi$ and each $i \in I$, all local absolute minima of the function $t \mapsto \langle \alpha_i^\vee, \eta(t) \rangle$ are integers.

We denote the set of all integral paths by $\Pi$. Obviously, $\Pi$ is a subcrystal of $\tilde{\Pi}$. Moreover, the general definition of the root operators in [33, Sect. 1] simplifies in the case of integral paths, which only need to be cut into three parts: the initial part is left invariant, the second part is reflected, and the third part is translated. Specifically, given $(\pi, \eta) \in \Pi^2$ and $i \in I$, we have $\eta = \tilde{e}_i \pi$ if and only if there exist a negative integer $p \in \mathbb{Z}$ and two reals $a$ and $b$ with $0 \le a < b \le 1$ such that the function $t \mapsto \langle \alpha_i^\vee, \pi(t) \rangle$ is weakly decreasing on $[a, b]$, and for each $t \in [0, 1]$,

* if $t \le a$, then $\langle \alpha_i^\vee, \pi(t) \rangle \ge p + 1$ and $\eta(t) = \pi(t)$;

* if $t = a$, then $\langle \alpha_i^\vee, \pi(t) \rangle = p + 1$;

* if $a < t < b$, then $p \le \langle \alpha_i^\vee, \pi(t) \rangle < p + 1$ and $\eta(t) = \pi(t) - (\langle \alpha_i^\vee, \pi(t) \rangle - p - 1)\alpha_i$;

* if $t = b$, then $\langle \alpha_i^\vee, \pi(t) \rangle = p$;

* if $t \ge b$, then $\langle \alpha_i^\vee, \pi(t) \rangle \ge p$ and $\eta(t) = \pi(t) + \alpha_i$.

We say that an integral path $\pi \in \Pi$ is *dominant* if its image is contained in $\Lambda_{\mathbb{R}}^+$.

**Remark 4.1.** Let $\Gamma$ be the group of all strictly increasing piecewise linear maps from $[0, 1]$ onto itself, the product being the composition of functions. This group acts on the set of all paths by right composition: $\pi \mapsto \pi \circ \gamma$ for a path $\pi$ and $\gamma \in \Gamma$. We say that $\pi \circ \gamma$ is obtained from $\pi$ by a *piecewise linear reparameterization*. Visibly, the set $\Pi$ of integral paths is stable under this action, the weight map $wt$ is invariant, and the root operators are equivariant. We can thus safely consider all our previous constructions modulo this action. We will sometimes implicitly assume that this quotient has been taken, i.e. among the possible parameterizations we choose one which is appropriate for the application in view.

The first two items in the following proposition ensure that there is an abundance of integral paths.

**Proposition 4.2.** (i) A dominant path $\pi$ is integral whenever for each $i \in I$, the function $t \mapsto \langle \alpha_i^\vee, \pi(t) \rangle$ is weakly increasing.

(ii) The set $\Pi$ is stable under concatenation of paths and the map $\pi \otimes \eta \mapsto \pi * \eta$ is a morphism of crystals from $\Pi \otimes \Pi$ to $\Pi$.

(iii) Let $\pi \in \Pi$. Then $A\pi$ contains a unique dominant path $\eta$ and is isomorphic as a crystal to $B(\mathrm{wt}(\eta))$.
structure on $\tilde{\Pi}$. Here the weight map is given by $wt(\pi) = \pi(1)$. To agree with the notation in Sect. 3.1, we will write $\tilde{e}_i$ and $\tilde{f}_i$ instead of $e_{\alpha_i}$ and $f_{\alpha_i}$ for each $i \in I$.

Let $\ell: [0, 1] \to \mathbb{R}$ be a piecewise linear function. We say that $p \in \mathbb{R}$ is a local absolute minimum of $\ell$ if there exists a compact interval $[a, b] \subseteq [0, 1]$ over which $\ell$ takes the value $p$, and there exists an $\epsilon > 0$ such that $\ell(x) > p$ for all $x \in (a - \epsilon, a) \cap [0, 1]$ and all $x \in (b, b + \epsilon) \cap [0, 1]$.

Given $\pi \in \tilde{\Pi}$, we denote by $A\pi$ the set of all paths $\eta \in \tilde{\Pi}$ that can be obtained from $\pi$ by applying a finite sequence of root operators $\tilde{e}_i$ or $\tilde{f}_i$. We say that a path $\pi \in \tilde{\Pi}$ is integral if for each $\eta \in A\pi$ and each $i \in I$, all local absolute minima of the function $t \mapsto \langle \alpha_i^\vee, \eta(t) \rangle$ are integers.

We denote the set of all integral paths by $\Pi$. Obviously, $\Pi$ is a subcrystal of $\tilde{\Pi}$. Moreover, the general definition of the root operators in [33, Sect. 1] simplifies in the case of integral paths, which only need to be cut into three parts: the initial part is left invariant, the second part is reflected, and the third part is translated. Specifically, given $(\pi, \eta) \in \Pi^2$ and $i \in I$, we have $\eta = \tilde{e}_i \pi$ if and only if there exist a negative integer $p \in \mathbb{Z}$ and two reals $a$ and $b$ with $0 \le a < b \le 1$ such that the function $t \mapsto \langle \alpha_i^\vee, \pi(t) \rangle$ is weakly decreasing on $[a, b]$, and for each $t \in [0, 1]$,

* if $t \le a$, then $\langle \alpha_i^\vee, \pi(t) \rangle \ge p + 1$ and $\eta(t) = \pi(t)$;

* if $t = a$, then $\langle \alpha_i^\vee, \pi(t) \rangle = p + 1$;

* if $a < t < b$, then $p \le \langle \alpha_i^\vee, \pi(t) \rangle < p + 1$ and $\eta(t) = \pi(t) - (\langle \alpha_i^\vee, \pi(t) \rangle - p - 1)\alpha_i$;

* if $t = b$, then $\langle \alpha_i^\vee, \pi(t) \rangle = p$;

* if $t \ge b$, then $\langle \alpha_i^\vee, \pi(t) \rangle \ge p$ and $\eta(t) = \pi(t) + \alpha_i$.

We say that an integral path $\pi \in \Pi$ is *dominant* if its image is contained in $\Lambda_{\mathbb{R}}^+$.

**Remark 4.1.** Let $\Gamma$ be the group of all strictly increasing piecewise linear maps from $[0, 1]$ onto itself, the product being the composition of functions. This group acts on the set of all paths by right composition: $\pi \mapsto \pi \circ \gamma$ for a path $\pi$ and $\gamma \in \Gamma$. We say that $\pi \circ \gamma$ is obtained from $\pi$ by a *piecewise linear reparameterization*. Visibly, the set $\Pi$ of integral paths is stable under this action, the weight map $wt$ is invariant, and the root operators are equivariant. We can thus safely consider all our previous constructions modulo this action. We will sometimes implicitly assume that this quotient has been taken, i.e. among the possible parameterizations we choose one which is appropriate for the application in view.

The first two items in the following proposition ensure that there is an abundance of integral paths.

**Proposition 4.2.** (i) A dominant path $\pi$ is integral whenever for each $i \in I$, the function $t \mapsto \langle \alpha_i^\vee, \pi(t) \rangle$ is weakly increasing.

(ii) The set $\Pi$ is stable under concatenation of paths and the map $\pi \otimes \eta \mapsto \pi * \eta$ is a morphism of crystals from $\Pi \otimes \Pi$ to $\Pi$.

(iii) Let $\pi \in \Pi$. Then $A\pi$ contains a unique dominant path $\eta$ and is isomorphic as a crystal to $B(\mathrm{wt}(\eta))$.
**Proposition 4.2** follows from the results in [35]. Two lemmas will help us bridge the gap.

We fix a scalar product $(\cdot, \cdot)$ on $\Lambda_R$ and let $d(\cdot, \cdot)$ be the corresponding distance function. We fix a basis $\mathbb{B}$ of $\Lambda$ and let $\mathbb{L}_1 \subset \Lambda_R$ be the associated unit cube, i.e. the set of points in $\Lambda_R$ which can be written as a linear combination of $\mathbb{B}$ with coefficients in the interval $[0, 1]$. Let $M$ be the maximal distance between two points in $\mathbb{L}_1$. Let $P \in \Lambda_Q^+$ be a dominant rational point and let $S(P, 1)$ be the sphere with center $P$ and radius $1$. Let $g$ be a ray starting at $P$ and let $g_1$ be the intersection point of this ray with the sphere $S(P, 1)$.

**Lemma 4.3.** For any $\epsilon > 0$ there is a ray $f$ starting at $P$ such that $(f \setminus \{P\}) \cap \Lambda \neq \emptyset$, and for $\{f_1\} = f \cap S(P, 1)$ we have $d(g_1, f_1) < \epsilon$.

*Proof.* Parameterize $g$ by $g(t) = P + t(g_1 - P)$ for $t \ge 0$. Choose $t_2 \gg 0$ and pick $\lambda \in \Lambda$ such that $g(t_2) \in \lambda + \mathbb{L}_1$. Let $f$ be the ray starting at $P$ passing through $\lambda$. Let $f_1$ be the intersection point of this ray with $S(P, 1)$; then $f(t) = P + t(f_1 - P)$ for $t \ge 0$ is a parameterization of $f$. Set $g_2 = g(t_2)$ and $f_2 = f(t_2)$. Noting that $d(P, f_2) = t_2 = d(P, g_2)$ and using the triangular inequality, we get

$$d(f_2, \lambda) = |d(P, f_2) - d(P, \lambda)| = |d(P, g_2) - d(P, \lambda)| \le d(g_2, \lambda) \le M,$$

whence $d(g_2, f_2) \le 2M$. By the intercept theorem $d(f_1, g_1)/d(g_2, f_2) = 1/t_2$, and hence $d(f_1, g_1) \le 2M/t_2$. For $t_2$ large enough we obtain $d(g_1, f_1) < \epsilon$. ■

In [35], a seemingly complicated definition of locally integral concatenation is introduced; it is a generalization of the concept of LS-paths [33]. This notion provides a sufficient condition for a path to be integral. Let us review it in the case of a rather special class of paths for which the property of being a locally integral concatenation reduces to condition (*) below.

We extend the concatenation operation $*$ to paths that do not necessarily end at an integral weight. For $\mu \in \Lambda_R$, let $\pi_\mu$ be the map $[0, 1] \to \Lambda_R$, $t \mapsto t\mu$. A path $\pi \in \tilde{\Pi}$ is said to be dominant rational if it is of the form $\pi = \pi_{\mu_1} * \cdots * \pi_{\mu_s}$, where $(\mu_1, \dots, \mu_s) \in (\Lambda_Q^+)^s$ and $\mu_1 + \cdots + \mu_s \in \Lambda^+$. For such a path, being a locally integral concatenation (see [35, Definition 5.3]) means:

(*) For each $j \in \{1, \dots, s\}$ such that $\mu_j \neq 0$, the affine line passing through $\mu_1 + \cdots + \mu_{j-1}$ and $\mu_1 + \cdots + \mu_j$ meets at least two lattice points.

An equivalent formulation: the affine line meets at least one rational point and one lattice point.

**Lemma 4.4.** A dominant rational path can be approximated by a locally integral concatenation.

*Proof.* Let $\pi = \pi_{\mu_1} * \cdots * \pi_{\mu_s}$ be a dominant rational path ending in $\mu \in \Lambda^+$. We define the support of an element $\mu \in \Lambda_R$ as the set of indices $i \in I$ such that $\langle \alpha_i^\vee, \mu \rangle \neq 0$. We can assume that the support of each $\mu_j$ is the same as the support of $\mu$; otherwise
**Proposition 4.2** follows from the results in [35]. Two lemmas will help us bridge the gap.

We fix a scalar product $(\cdot, \cdot)$ on $\Lambda_R$ and let $d(\cdot, \cdot)$ be the corresponding distance function. We fix a basis $\mathbb{B}$ of $\Lambda$ and let $\mathbb{L}_1 \subset \Lambda_R$ be the associated unit cube, i.e. the set of points in $\Lambda_R$ which can be written as a linear combination of $\mathbb{B}$ with coefficients in the interval $[0, 1]$. Let $M$ be the maximal distance between two points in $\mathbb{L}_1$. Let $P \in \Lambda_Q^+$ be a dominant rational point and let $S(P, 1)$ be the sphere with center $P$ and radius 1. Let $g$ be a ray starting at $P$ and let $g_1$ be the intersection point of this ray with the sphere $S(P, 1)$.

**Lemma 4.3.** For any $\epsilon > 0$ there is a ray $f$ starting at $P$ such that $(f \setminus \{P\}) \cap \Lambda \neq \emptyset$, and for $\{f_1\} = f \cap S(P, 1)$ we have $d(g_1, f_1) < \epsilon$.

*Proof.* Parameterize $g$ by $g(t) = P + t(g_1 - P)$ for $t \ge 0$. Choose $t_2 \gg 0$ and pick $\lambda \in \Lambda$ such that $g(t_2) \in \lambda + \mathbb{L}_1$. Let $f$ be the ray starting at $P$ passing through $\lambda$. Let $f_1$ be the intersection point of this ray with $S(P, 1)$; then $f(t) = P + t(f_1 - P)$ for $t \ge 0$ is a parameterization of $f$. Set $g_2 = g(t_2)$ and $f_2 = f(t_2)$. Noting that $d(P, f_2) = t_2 = d(P, g_2)$ and using the triangular inequality, we get

$$d(f_2, \lambda) = |d(P, f_2) - d(P, \lambda)| = |d(P, g_2) - d(P, \lambda)| \le d(g_2, \lambda) \le M,$$

whence $d(g_2, f_2) \le 2M$. By the intercept theorem $d(f_1, g_1)/d(g_2, f_2) = 1/t_2$, and hence $d(f_1, g_1) \le 2M/t_2$. For $t_2$ large enough we obtain $d(g_1, f_1) < \epsilon$. ■

In [35], a seemingly complicated definition of locally integral concatenation is introduced; it is a generalization of the concept of LS-paths [33]. This notion provides a sufficient condition for a path to be integral. Let us review it in the case of a rather special class of paths for which the property of being a locally integral concatenation reduces to condition (*) below.

We extend the concatenation operation $*$ to paths that do not necessarily end at an integral weight. For $\mu \in \Lambda_R$, let $\pi_\mu$ be the map $[0, 1] \to \Lambda_R$, $t \mapsto t\mu$. A path $\pi \in \tilde{\Pi}$ is said to be dominant rational if it is of the form $\pi = \pi_{\mu_1} * \cdots * \pi_{\mu_s}$, where $(\mu_1, \dots, \mu_s) \in (\Lambda_Q^+)^s$ and $\mu_1 + \cdots + \mu_s \in \Lambda^+$. For such a path, being a locally integral concatenation (see [35, Definition 5.3]) means:

(*) For each $j \in \{1, \dots, s\}$ such that $\mu_j \neq 0$, the affine line passing through $\mu_1 + \cdots + \mu_{j-1}$ and $\mu_1 + \cdots + \mu_j$ meets at least two lattice points.

An equivalent formulation: the affine line meets at least one rational point and one lattice point.

**Lemma 4.4.** A dominant rational path can be approximated by a locally integral concatenation.

*Proof.* Let $\pi = \pi_{\mu_1} * \cdots * \pi_{\mu_s}$ be a dominant rational path ending in $\mu \in \Lambda^+$. We define the support of an element $\mu \in \Lambda_R$ as the set of indices $i \in I$ such that $\langle \alpha_i^\vee, \mu \rangle \neq 0$. We can assume that the support of each $\mu_j$ is the same as the support of $\mu$; otherwise
we approximate $\pi$ by a path we get by slightly perturbing $\mu_1, \dots, \mu_s$, for instance by replacing each $\mu_j$ by $\mu_j + \epsilon(\mu/s - \mu_j)$ for some rational $0 < \epsilon \ll 1$. We can also assume the support of $\mu$ is $I$, otherwise we work within the subspace $\bigcap_{i \in I \setminus \text{supp}(\mu)} \text{ker } \alpha_i^{\vee}$.

Under these assumptions, each weight $\mu_j$ is regular dominant. Then small perturbations $\mu'_1, \dots, \mu'_{s-1}$ of the directions $\mu_1, \dots, \mu_{s-1}$ remain dominant, and so does $\mu'_s = \mu - (\mu'_1 + \dots + \mu'_{s-1})$. By Lemma 4.3, one can perturb in such a way that the new path $\eta := \pi_{\mu'_1} * \dots * \pi_{\mu'_s}$ is a dominant rational path and the first $s-1$ line segments of $\eta$ satisfy the affine line condition $(*)$. The last line segment of $\eta$ meets the lattice point $\mu$ and a rational point, and thus satisfies $(*)$ too. Hence $\eta$ is a locally integral concatenation, approximating the dominant rational path $\pi$. ■

**Proof of Proposition 4.2.** A path in statement (i) is of the form $\pi = \pi_{\mu_1} * \dots * \pi_{\mu_s}$, where $(\mu_1, \dots, \mu_s) \in (\Lambda_{\mathbb{R}}^{\vee})^s$ are dominant. Such a path can be approximated by a dominant rational path without altering $\mu_1 + \dots + \mu_s$. In turn, a rational dominant path can be approximated by a locally integral concatenation by Lemma 4.4. Lastly, a locally integral concatenation is an integral path by [35, Lemma 5.6] and Proposition 5.9. The integrality property in (i) follows now by the continuity of the root operators [35, property (v) CONTINUITY].

It remains to prove the other two statements. The endpoint of a path is by definition an element of the lattice, so the concatenation of integral paths is an integral path. Moreover, by [35, Lemma 6.12], concatenation defines a morphism of crystals $\Pi \otimes \Pi \to \Pi$. This shows (ii). Statement (iii) follows by [35, Lemma 6.11]. ■

## 4.2. From paths to MV cycles

We need additional terminology before we proceed to the main construction of this section.

To each coroot $\alpha^{\vee} \in \Phi^{\vee}$ corresponds an additive one-parameter subgroup $x_{\alpha^{\vee}} : \mathbb{G}_{\alpha} \to \mathcal{G}^{\vee}$. Given additionally an integer $p \in \mathbb{Z}$, we define a map

$$x_{(\alpha^{\vee}, p)} : \mathbb{C} \to \mathcal{G}^{\vee}(\mathcal{K}), \quad a \mapsto x_{\alpha^{\vee}}(az^p).$$

An affine coroot is a pair $(\alpha^{\vee}, p)$ consisting of a coroot $\alpha^{\vee} \in \Phi^{\vee}$ and an integer $p$. The direction of an affine coroot $(\alpha^{\vee}, p)$ is $\alpha^{\vee}$. An affine coroot is said to be positive if its direction is so. We denote the set of affine coroots by $\Phi_a^{\vee}$ and the set of positive affine coroots by $\Phi_a^{\vee,+}$.

To an affine coroot $\beta$, besides the map $x_{\beta}$ defined above, we attach a hyperplane $H_{\beta}$ and a negative closed half-space $H_{\beta}^{-}$ in $\Lambda_{\mathbb{R}}$ as follows:

$$H_{(\alpha^{\vee}, p)} = \{x \in \Lambda_{\mathbb{R}} \mid \langle \alpha^{\vee}, x \rangle = p\}, \quad H_{(\alpha^{\vee}, p)}^{-} = \{x \in \Lambda_{\mathbb{R}} \mid \langle \alpha^{\vee}, x \rangle \le p\}.$$

Let $s_{\beta}$ be the reflection across the hyperplane $H_{\beta}$; concretely,

$$s_{(\alpha^{\vee}, p)}(x) = x - (\langle \alpha^{\vee}, x \rangle - p)\alpha$$

for any $x \in \Lambda_{\mathbb{R}}$. In addition, we denote by $\tau_{\lambda}$ the translation $x \mapsto x + \lambda$ by the element $\lambda \in \Lambda$. The subgroup of Aut($\Lambda_{\mathbb{R}}$) generated by all the reflections $s_{\beta}$ is the affine Weyl
we approximate $\pi$ by a path we get by slightly perturbing $\mu_1, \dots, \mu_s$, for instance by
replacing each $\mu_j$ by $\mu_j + \epsilon(\mu/s - \mu_j)$ for some rational $0 < \epsilon \ll 1$. We can also assume
the support of $\mu$ is $I$, otherwise we work within the subspace $\bigcap_{i \in I \setminus \text{supp}(\mu)} \text{ker } \alpha_i^\vee$.

Under these assumptions, each weight $\mu_j$ is regular dominant. Then small perturba-
tions $\mu'_1, \dots, \mu'_{s-1}$ of the directions $\mu_1, \dots, \mu_{s-1}$ remain dominant, and so does $\mu'_s =$
$\mu - (\mu'_1 + \dots + \mu'_{s-1})$. By Lemma 4.3, one can perturb in such a way that the new path
$\eta := \pi_{\mu'_1} * \dots * \pi_{\mu'_s}$ is a dominant rational path and the first $s-1$ line segments of $\eta$
satisfy the affine line condition $(*)$. The last line segment of $\eta$ meets the lattice point $\mu$
and a rational point, and thus satisfies $(*)$ too. Hence $\eta$ is a locally integral concatenation,
approximating the dominant rational path $\pi$.

■

*Proof of Proposition 4.2.* A path in statement (i) is of the form $\pi = \pi_{\mu_1} * \dots * \pi_{\mu_s}$, where $(\mu_1, \dots, \mu_s) \in (\Lambda_{\mathbb{R}}^{\vee})^s$ are dominant. Such a path can be approximated by a dominant rational path without altering $\mu_1 + \dots + \mu_s$. In turn, a rational dominant path can be approximated by a locally integral concatenation by Lemma 4.4. Lastly, a locally integral concatenation is an integral path by [35, Lemma 5.6] and Proposition 5.9. The integrality property in (i) follows now by the continuity of the root operators [35, property (v) CONTINUITY].

It remains to prove the other two statements. The endpoint of a path is by definition an
element of the lattice, so the concatenation of integral paths is an integral path. Moreover,
by [35, Lemma 6.12], concatenation defines a morphism of crystals $\Pi \otimes \Pi \to \Pi$. This
shows (ii). Statement (iii) follows by [35, Lemma 6.11].

■

## 4.2. *From paths to MV cycles*

We need additional terminology before we proceed to the main construction of this sec-
tion.

To each coroot $\alpha^{\vee} \in \Phi^{\vee}$ corresponds an additive one-parameter subgroup $x_{\alpha^{\vee}} : \mathbb{G}_{\alpha} \to \mathcal{G}^{\vee}$. Given additionally an integer $p \in \mathbb{Z}$, we define a map

$$
x_{(\alpha^{\vee}, p)} : \mathbb{C} \to \mathcal{G}^{\vee}(\mathcal{K}), \quad a \mapsto x_{\alpha^{\vee}}(az^p).
$$

An *affine coroot* is a pair $(\alpha^{\vee}, p)$ consisting of a coroot $\alpha^{\vee} \in \Phi^{\vee}$ and an integer $p$.
The *direction* of an affine coroot $(\alpha^{\vee}, p)$ is $\alpha^{\vee}$. An affine coroot is said to be *positive* if
its direction is so. We denote the set of affine coroots by $\Phi_{\alpha}^{\vee}$ and the set of positive affine
coroots by $\Phi_{\alpha}^{\vee,+}$.

To an affine coroot $\beta$, besides the map $x_\beta$ defined above, we attach a hyperplane $H_\beta$
and a negative closed half-space $H_\beta^-$ in $\Lambda_\mathbb{R}$ as follows:

$$
H_{(\alpha^{\vee}, p)} = \{x \in \Lambda_{\mathbb{R}} \mid \langle \alpha^{\vee}, x \rangle = p\}, \quad H_{(\alpha^{\vee}, p)}^{-} = \{x \in \Lambda_{\mathbb{R}} \mid \langle \alpha^{\vee}, x \rangle \le p\}.
$$

Let $s_\beta$ be the reflection across the hyperplane $H_\beta$; concretely,

$$
s_{(\alpha^{\vee}, p)}(x) = x - (\langle \alpha^{\vee}, x \rangle - p)\alpha
$$

for any $x \in \Lambda_{\mathbb{R}}$. In addition, we denote by $\tau_{\lambda}$ the translation $x \mapsto x + \lambda$ by the element
$\lambda \in \Lambda$. The subgroup of Aut($\Lambda_{\mathbb{R}}$) generated by all the reflections $s_{\beta}$ is the affine Weyl
group $W_a$. When we add the translations $\tau_\lambda$, we obtain the extended affine Weyl group $\tilde{W}_a$. Then $\tau_\lambda \in W_a$ if and only if $\lambda \in \mathbb{Z}\Phi$.

The group $\tilde{W}_a$ acts on the set $\Phi_a^\vee$ of affine roots: one demands that $w(H_\beta^-) = H_{w\beta}$ for each element $w \in \tilde{W}_a$ and each affine coroot $\beta \in \Phi_a^\vee$. Then for each $\beta \in \Phi_a^\vee$ and each $\lambda \in \Lambda$, we have $x_{\tau_\lambda \beta}(a) = z^\lambda x_\beta(a)z^{-\lambda}$ for all $a \in \mathbb{C}$.

We denote by $\mathfrak{S}$ the arrangement formed by the hyperplanes $H_\beta$, where $\beta \in \Phi_a^\vee$. It divides the vector space $\Lambda_\mathbb{R}$ into faces. The closure of a face is the disjoint union of faces of smaller dimension. Endowed with the set of all faces, $\Lambda_\mathbb{R}$ becomes a polysimplicial complex, called the *affine Coxeter complex*.

For each face $f$ of the affine Coxeter complex, we denote by $N^\vee(f)$ the subgroup of $N^\vee(\mathcal{K})$ generated by the elements $x_{\alpha^\vee}(az^p)$, where $a \in \mathcal{O}$ and $(\alpha^\vee, p)$ is a positive affine coroot such that $f \subseteq H_{(\alpha^\vee, p)}$. We note that $N^\vee(\tau_\lambda f) = z^\lambda N^\vee(f)z^{-\lambda}$ for each face $f$ and each $\lambda \in \Lambda$ and that $N^\vee(f) \subseteq N^\vee(\mathcal{O})$ if $f \subseteq \Lambda_\mathbb{R}^+$.

For $x \in \Lambda_\mathbb{R}$, we denote by $f_x$ the face in the affine Coxeter complex that contains the point $x$. We use the symbol $\prod'$ to denote the restricted product of groups, consisting of families involving only finitely many nontrivial terms.

Using these conventions and the notation introduced in Sect. 2.2, given $(\pi_1, \dots, \pi_n) \in \Pi^n$, we define $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ as the subset of $\mathrm{Gr}_n$ of all elements

$$ \left[ \left( \prod_{t_1 \in [0,1]} v_{1,t_1} \right) z^{\mathrm{wt}(\pi_1)}, \dots, \left( \prod_{t_n \in [0,1]} v_{n,t_n} \right) z^{\mathrm{wt}(\pi_n)} \right] $$

with

$$ ((v_{1,t_1}), \dots, (v_{n,t_n})) \in \prod'_{t_1 \in [0,1]} N^\vee(f_{\pi_1(t_1)}) \times \dots \times \prod'_{t_n \in [0,1]} N^\vee(f_{\pi_n(t_n)}). $$

**Proposition 4.5.** Let $(\pi_1, \dots, \pi_n) \in \Pi^n$.

(i) The set $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is stable under left multiplication by $N^\vee(\mathcal{O})$.

(ii) Let $\mu = \mathrm{wt}(\pi_1) + \dots + \mathrm{wt}(\pi_n)$. Then the set $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is an irreducible constructible subset of $(m_n)^{-1}(S_\mu)$.

(iii) We have

$$ \begin{aligned} \mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n) &= \Psi(\mathring{Z}(\pi_1) \ltimes \dots \ltimes \mathring{Z}(\pi_n)), \\ \mathring{Z}(\pi_1 * \dots * \pi_n) &= m_n(\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)). \end{aligned} $$

(iv) Let $i \in I$ and compute $\eta_1 \otimes \dots \otimes \eta_n = \tilde{e}_i(\pi_1 \otimes \dots \otimes \pi_n)$ in the crystal $\Pi^{\otimes n}$, provided that this operation is defined. Then $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is contained in the closure of $\mathring{Z}(\eta_1 \otimes \dots \otimes \eta_n)$ in $\mathrm{Gr}_n$.

*Proof.* Assertion (i) is a direct consequence of the equality $N^\vee(f_{\pi_1(0)}) = N^\vee(f_0) = N^\vee(\mathcal{O})$.

Assertion (ii) comes from general principles once we have replaced the restricted infinite product by a finite one.
group $W_a$. When we add the translations $\tau_\lambda$, we obtain the extended affine Weyl group $\tilde{W}_a$. Then $\tau_\lambda \in W_a$ if and only if $\lambda \in \mathbb{Z}\Phi$.

The group $\tilde{W}_a$ acts on the set $\Phi_a^\vee$ of affine roots: one demands that $w(H_\beta^-) = H_{w\beta}^-$ for each element $w \in \tilde{W}_a$ and each affine coroot $\beta \in \Phi_a^\vee$. Then for each $\beta \in \Phi_a^\vee$ and each $\lambda \in \Lambda$, we have $x_{\tau_\lambda\beta}(a) = z^\lambda x_\beta(a)z^{-\lambda}$ for all $a \in \mathbb{C}$.

We denote by $\mathfrak{S}$ the arrangement formed by the hyperplanes $H_\beta$, where $\beta \in \Phi_a^\vee$. It divides the vector space $\Lambda_\mathbb{R}$ into faces. The closure of a face is the disjoint union of faces of smaller dimension. Endowed with the set of all faces, $\Lambda_\mathbb{R}$ becomes a polysimplicial complex, called the *affine Coxeter complex*.

For each face $f$ of the affine Coxeter complex, we denote by $N^\vee(f)$ the subgroup of $N^\vee(\mathcal{K})$ generated by the elements $x_{\alpha^\vee}(az^p)$, where $a \in \mathcal{O}$ and $(\alpha^\vee, p)$ is a positive affine coroot such that $f \subseteq H_{(\alpha^\vee, p)}$. We note that $N^\vee(\tau_\lambda f) = z^\lambda N^\vee(f)z^{-\lambda}$ for each face $f$ and each $\lambda \in \Lambda$ and that $N^\vee(f) \subseteq N^\vee(\mathcal{O})$ if $f \subseteq \Lambda_\mathbb{R}^+$.

For $x \in \Lambda_\mathbb{R}$, we denote by $f_x$ the face in the affine Coxeter complex that contains the point $x$. We use the symbol $\prod'$ to denote the restricted product of groups, consisting of families involving only finitely many nontrivial terms.

Using these conventions and the notation introduced in Sect. 2.2, given $(\pi_1, \dots, \pi_n) \in \Pi^n$, we define $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ as the subset of $\mathrm{Gr}_n$ of all elements

$$ \left[ \left( \prod_{t_1 \in [0,1]} v_{1,t_1} \right) z^{\mathrm{wt}(\pi_1)}, \dots, \left( \prod_{t_n \in [0,1]} v_{n,t_n} \right) z^{\mathrm{wt}(\pi_n)} \right] $$

with

$$ ((v_{1,t_1}), \dots, (v_{n,t_n})) \in \prod'_{t_1 \in [0,1]} N^\vee(f_{\pi_1(t_1)}) \times \dots \times \prod'_{t_n \in [0,1]} N^\vee(f_{\pi_n(t_n)}). $$

**Proposition 4.5.** Let $(\pi_1, \dots, \pi_n) \in \Pi^n$.

(i) The set $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is stable under left multiplication by $N^\vee(\mathcal{O})$.

(ii) Let $\mu = \mathrm{wt}(\pi_1) + \dots + \mathrm{wt}(\pi_n)$. Then the set $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is an irreducible constructible subset of $(m_n)^{-1}(S_\mu)$.

(iii) We have

$$ \begin{aligned} \mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n) &= \Psi(\mathring{Z}(\pi_1) \ltimes \dots \ltimes \mathring{Z}(\pi_n)), \\ \mathring{Z}(\pi_1 * \dots * \pi_n) &= m_n(\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)). \end{aligned} $$

(iv) Let $i \in I$ and compute $\eta_1 \otimes \dots \otimes \eta_n = \tilde{e}_i(\pi_1 \otimes \dots \otimes \pi_n)$ in the crystal $\Pi^{\otimes n}$, provided that this operation is defined. Then $\mathring{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is contained in the closure of $\mathring{Z}(\eta_1 \otimes \dots \otimes \eta_n)$ in $\mathrm{Gr}_n$.

*Proof.* Assertion (i) is a direct consequence of the equality $N^\vee(f_{\pi_1(0)}) = N^\vee(f_0) = N^\vee(\mathcal{O})$.

Assertion (ii) comes from general principles once we have replaced the restricted infinite product by a finite one.
The first equation in (iii) is tautological. In the second one, we view the concatenation $\pi = \pi_1 * \cdots * \pi_n$ as a map from $[0, n]$ to $\Lambda_{\mathbb{R}}$, each path $\pi_1, \dots, \pi_n$ being travelled at nominal speed. We set $v_0 = 0$, and for $j \in \{2, \dots, n\}$ we set $v_{j-1} = \text{wt}(\pi_1) + \cdots + \text{wt}(\pi_{j-1})$. Then for $j \in \{1, \dots, n\}$ and $t \in [0, 1]$, we have $\pi(t + (j-1)) = v_{j-1} + \pi_j(t)$, and accordingly $N^\vee(\mathfrak{f}_{\pi(t+(j-1))}) = z^{v_{j-1}} N^\vee(\mathfrak{f}_{\pi_j(t)})z^{-v_{j-1}}$. A banal calculation then yields the desired result.

The proof of (iv) is much more involved. We defer its presentation to Sect. 4.5. ■

For $(\pi_1, \dots, \pi_n) \in \Pi^n$, we denote by $\mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ the closure of $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ in $(m_n)^{-1}(S_\mu)$, where $\mu = \text{wt}(\pi_1) + \cdots + \text{wt}(\pi_n)$.

**Theorem 4.6.** Let $(\pi_1, \dots, \pi_n) \in \Pi^n$, set $\mu = \text{wt}(\pi_1) + \cdots + \text{wt}(\pi_n)$ and for $j \in \{1, \dots, n\}$, let $\lambda_j$ be the weight of the unique dominant path in $A\pi_j$. Then $\mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ is an MV cycle; specifically, $\mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n) \in_* \mathcal{L}(\lambda_1, \dots, \lambda_n)_\mu$.

*Proof.* We start with the particular case $n=1$. Let $\pi \in \Pi$, let $\eta$ be the unique dominant path in $A\pi$, set $\lambda = \text{wt}(\eta)$ and $\mu = \text{wt}(\pi)$, and set $p = 2\rho(\lambda)$ and $k = \rho(\lambda + \mu)$. By Proposition 4.2 (iii), the crystal $A\pi$ is isomorphic to $B(\lambda)$, so it contains a unique lowest weight element $\xi$. Then $\pi$ can be reached by applying a sequence of root operators $\tilde{f}_i$ to $\eta$ or by applying a sequence of root operators $\tilde{e}_i$ to $\xi$. Thus, there exists a finite sequence $(\pi_0, \dots, \pi_p)$ of elements in $A\pi$ such that $\pi_0 = \xi$, $\pi_k = \pi$, $\pi_p = \eta$ and each $\pi_{j+1}$ is obtained from $\pi_j$ by applying a root operator $\tilde{e}_i$.

Since $\eta$ is dominant, each face $\mathfrak{f}_{\eta(t)}$ is contained in $\Lambda_{\mathbb{R}}^+$, so each group $N^\vee(\mathfrak{f}_{\eta(t)})$ is contained in $N^\vee(\mathcal{O})$. Then by construction $\mathring{\mathbb{Z}}(\eta) \subseteq N^\vee(\mathcal{O})L_\lambda$, and therefore $\overline{\mathbb{Z}}(\eta)$ (the closure of $\mathbb{Z}(\eta)$ in $\text{Gr}$) is contained in $\overline{\text{Gr}}^\lambda$. Also, Proposition 4.5 (iv) implies that

$$ \overline{\mathbb{Z}}(\pi_0) \subseteq \overline{\mathbb{Z}}(\pi_1) \subseteq \cdots \subseteq \overline{\mathbb{Z}}(\pi_p). \quad (14) $$

These inclusions are strict because $\overline{\mathbb{Z}}(\pi_j)$ is contained in the closure of $\overline{\text{Gr}}^\lambda \cap S_{\text{wt}(\pi_j)}$, which is disjoint from $S_{\text{wt}(\pi_{j+1})}$ by [39, Proposition 3.1 (a)], while $\mathbb{Z}(\pi_{j+1})$ is contained in $S_{\text{wt}(\pi_{j+1})}$. Thus (14) is a strictly increasing chain of closed irreducible subsets of $\overline{\text{Gr}}^\lambda$. As $\overline{\text{Gr}}^\lambda$ has dimension $p$, we see that each $\overline{\mathbb{Z}}(\pi_j)$ has dimension $j$.

In particular, $\overline{\mathbb{Z}}(\pi)$ has dimension $k$. But $\mathbb{Z}(\pi)$ is locally closed, because it is a closed subset of $S_\mu$ which is locally closed. So $\mathbb{Z}(\pi)$ has dimension $k$. At this point, we know that $\mathbb{Z}(\pi)$ is a closed irreducible subset of $\overline{\text{Gr}}^\lambda \cap S_\mu$ of dimension $k = \rho(\lambda + \mu)$. Therefore $\mathbb{Z}(\pi)$ belongs to $\mathcal{L}(\lambda)_\mu$.

The reasoning above establishes the case $n=1$ of the theorem. The general case then follows from Propositions 2.2 (ii) and 4.5 (iii). ■

### 4.3. A more economical definition

For the proofs in the following sections, it will be convenient to have a more economical presentation of the sets $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$. We need a few additional pieces of notation.
The first equation in (iii) is tautological. In the second one, we view the concatenation
$\pi = \pi_1 * \cdots * \pi_n$ as a map from $[0, n]$ to $\Lambda_{\mathbb{R}}$, each path $\pi_1, \dots, \pi_n$ being travelled
at nominal speed. We set $v_0 = 0$, and for $j \in \{2, \dots, n\}$ we set $v_{j-1} = \text{wt}(\pi_1) + \cdots +$
$\text{wt}(\pi_{j-1})$. Then for $j \in \{1, \dots, n\}$ and $t \in [0, 1]$, we have $\pi(t + (j-1)) = v_{j-1} + \pi_j(t)$,
and accordingly $N^\vee(\mathfrak{f}_{\pi(t+(j-1))}) = z^{v_{j-1}} N^\vee(\mathfrak{f}_{\pi_j(t)})z^{-v_{j-1}}$. A banal calculation then
yields the desired result.

The proof of (iv) is much more involved. We defer its presentation to Sect. 4.5.

For $(\pi_1, \dots, \pi_n) \in \Pi^n$, we denote by $\mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ the closure of $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ in $(m_n)^{-1}(S_\mu)$, where $\mu = \text{wt}(\pi_1) + \cdots + \text{wt}(\pi_n)$.

**Theorem 4.6.** Let $(\pi_1, \dots, \pi_n) \in \Pi^n$, set $\mu = \text{wt}(\pi_1) + \dots + \text{wt}(\pi_n)$ and for $j \in \{1, \dots, n\}$, let $\lambda_j$ be the weight of the unique dominant path in $A\pi_j$. Then $\mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ is an MV cycle; specifically, $\mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n) \in_* \mathcal{L}(\lambda_1, \dots, \lambda_n)_\mu$.

*Proof.* We start with the particular case $n=1$. Let $\pi \in \Pi$, let $\eta$ be the unique dominant path in $A\pi$, set $\lambda = \text{wt}(\eta)$ and $\mu = \text{wt}(\pi)$, and set $p = 2\rho(\lambda)$ and $k = \rho(\lambda + \mu)$. By Proposition 4.2 (iii), the crystal $A\pi$ is isomorphic to $B(\lambda)$, so it contains a unique lowest weight element $\xi$. Then $\pi$ can be reached by applying a sequence of root operators $\tilde{f}_i$ to $\eta$ or by applying a sequence of root operators $\tilde{e}_i$ to $\xi$. Thus, there exists a finite sequence $(\pi_0, \dots, \pi_p)$ of elements in $A\pi$ such that $\pi_0 = \xi$, $\pi_k = \pi$, $\pi_p = \eta$ and each $\pi_{j+1}$ is obtained from $\pi_j$ by applying a root operator $\tilde{e}_i$.

Since $\eta$ is dominant, each face $\mathfrak{f}_{\eta(t)}$ is contained in $\Lambda_{\mathbb{R}}^{+}$, so each group $N^{\vee}(\mathfrak{f}_{\eta(t)})$ is contained in $N^{\vee}(\mathcal{O})$. Then by construction $\mathring{\mathbb{Z}}(\eta) \subseteq N^{\vee}(\mathcal{O})L_{\lambda}$, and therefore $\overline{\mathbb{Z}}(\eta)$ (the closure of $\mathbb{Z}(\eta)$ in $\text{Gr}$) is contained in $\overline{\text{Gr}}^{\lambda}$. Also, Proposition 4.5 (iv) implies that

$$
\overline{\mathbb{Z}(\pi_0)} \subseteq \overline{\mathbb{Z}(\pi_1)} \subseteq \dots \subseteq \overline{\mathbb{Z}(\pi_p)}. \qquad (14)
$$

These inclusions are strict because $\overline{\mathbb{Z}(\pi_j)}$ is contained in the closure of $\overline{\text{Gr}}^{\lambda} \cap S_{\text{wt}(\pi_j)}$,
which is disjoint from $S_{\text{wt}(\pi_{j+1})}$ by [39, Proposition 3.1 (a)], while $\mathbb{Z}(\pi_{j+1})$ is contained
in $S_{\text{wt}(\pi_{j+1})}$. Thus (14) is a strictly increasing chain of closed irreducible subsets of $\overline{\text{Gr}}^{\lambda}$.
As $\overline{\text{Gr}}^{\lambda}$ has dimension $p$, we see that each $\overline{\mathbb{Z}(\pi_j)}$ has dimension $j$.

In particular, $\overline{\mathbb{Z}}(\pi)$ has dimension $k$. But $\mathbb{Z}(\pi)$ is locally closed, because it is a closed subset of $S_\mu$ which is locally closed. So $\mathbb{Z}(\pi)$ has dimension $k$. At this point, we know that $\mathbb{Z}(\pi)$ is a closed irreducible subset of $\overline{\text{Gr}}^\lambda \cap S_\mu$ of dimension $k = \rho(\lambda + \mu)$. Therefore $\mathbb{Z}(\pi)$ belongs to $\mathcal{L}(\lambda)_\mu$.

The reasoning above establishes the case $n=1$ of the theorem. The general case then
follows from Propositions 2.2 (ii) and 4.5 (iii). ■

4.3. *A more economical definition*

For the proofs in the following sections, it will be convenient to have a more economical
presentation of the sets $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$. We need a few additional pieces of notation.
When $\mathfrak{f}$ and $\mathfrak{f}'$ are two faces of the affine Coxeter complex such that $\mathfrak{f}$ is contained in the closure $\bar{\mathfrak{f}}'$ of $\mathfrak{f}'$, we denote by $\Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')$ the set of all positive affine coroots $\beta$ such that $\mathfrak{f} \subseteq H_\beta$ and $\mathfrak{f}' \not\subseteq H_\beta^-$. We denote by $\mathcal{N}^\vee(\mathfrak{f}, \mathfrak{f}')$ the subgroup of $\mathcal{N}^\vee(\mathcal{K})$ generated by the elements $x_\beta(a)$ with $\beta \in \Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')$ and $a \in \mathbb{C}$.

We recall that a group $\Gamma$ is the (internal) Zappa–Szép product of two subgroups $\Gamma'$ and $\Gamma''$ if the product in $\Gamma$ induces a bijection $\Gamma' \times \Gamma'' \to \Gamma$. We indicate this situation with the notation $\Gamma = \Gamma' \bowtie \Gamma''$.

The following result is Proposition 19(ii) in [2].

**Lemma 4.7.** Let $\mathfrak{f}$ and $\mathfrak{f}'$ be two faces of the affine Coxeter complex such that $\mathfrak{f} \subseteq \bar{\mathfrak{f}}'$. Then $\mathcal{N}^\vee(\mathfrak{f}) = \mathcal{N}^\vee(\mathfrak{f}, \mathfrak{f}') \bowtie \mathcal{N}^\vee(\mathfrak{f}')$, and the map

$$ \mathbb{C}^{\Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}') \to \mathcal{N}^{\vee}(\mathfrak{f}, \mathfrak{f}')}, \quad (a_\beta) \mapsto \prod_{\beta \in \Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')} x_\beta(a_\beta), $$

is bijective, whichever order on $\Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')$ is used to compute the product.

Given $\pi \in \Pi$ and $t \in [0, 1[$, we denote by $\mathfrak{f}_{\pi(t+0)}$ the face in the affine Coxeter complex that contains the points $\pi(t+h)$ for all small enough $h > 0$. Obviously, its closure meets, hence contains, the face $\mathfrak{f}_{\pi(t)}$. We set $\Phi_a^{\mathfrak{f},+}(\pi, t) = \Phi_a^{\mathfrak{f},+}(\mathfrak{f}_{\pi(t)}, \mathfrak{f}_{\pi(t+0)})$ and $\mathcal{N}^\vee(\pi, t) = \mathcal{N}^\vee(\mathfrak{f}_{\pi(t)}, \mathfrak{f}_{\pi(t+0)})$.

Concretely, $\Phi_a^{\mathfrak{f},+}(\pi, t)$ is the set of all $\beta \in \Phi_a^{\mathfrak{f},+}$ such that $\pi$ quits the half-space $H_\beta^-$ at time $t$ and $\mathcal{N}^\vee(\pi, t)$ is the subgroup of $\mathcal{N}^\vee(\mathcal{K})$ generated by the elements $x_\beta(a)$ with $\beta \in \Phi_a^{\mathfrak{f},+}(\pi, t)$ and $a \in \mathbb{C}$. Note that $\Phi_a^{\mathfrak{f},+}(\pi, t)$ is empty save for finitely many $t$.

**Proposition 4.8.** Let $(\pi_1, \dots, \pi_n) \in \Pi^n$. Then $\mathring{\mathbb{O}}(\pi_1 \otimes \dots \otimes \pi_n)$ is the set of all elements

$$ \left[ \left( \prod_{t_1 \in [0,1[} v_{1,t_1} \right) z^{\mathrm{wt}(\pi_1)}, \dots, \left( \prod_{t_n \in [0,1[} v_{n,t_n} \right) z^{\mathrm{wt}(\pi_n)} \right] $$

with

$$ ((v_{1,t_1}, \dots, v_{n,t_n}) \in \prod_{t_1 \in [0,1[} \mathcal{N}^{\vee}(\pi_1, t_1) \times \dots \times \prod_{t_n \in [0,1[} \mathcal{N}^{\vee}(\pi_n, t_n)). $$

*Proof.* Let $\pi \in \Pi$. Let $(t_1, \dots, t_m)$ be the ordered list of all elements $t \in [0, 1[$ such that $\Phi_a^{\mathfrak{f},+}(\pi, t) \neq \emptyset$ and set $t_{m+1} = 1$.

Pick $\ell \in \{1, \dots, m\}$; between the times $t_\ell$ and $t_{\ell+1}$, the path $\pi$ never quits the half-space $H_\beta^-$ of a positive affine coroot $\beta$; as a consequence, the map $t \mapsto \mathcal{N}^\vee(\mathfrak{f}_{\pi(t)})$ is nondecreasing on $]t_\ell, t_{\ell+1}]$. It is also nondecreasing on $[0, t_1]$ if $t_1 > 0$. However, when $t$ goes past the point $t_\ell$, the group $\mathcal{N}^\vee(\mathfrak{f}_{\pi(t)})$ loses the first factor of the Zappa–Szép product $\mathcal{N}^\vee(\mathfrak{f}_{\pi(t_\ell)}) = \mathcal{N}^\vee(\pi, t_\ell) \bowtie \mathcal{N}^\vee(\mathfrak{f}_{\pi(t_\ell+0)})$.

It follows that for any family $(u_t)$ in $\prod'_{t \in [t_\ell, t_{\ell+1}]} \mathcal{N}^\vee(\mathfrak{f}_{\pi(t)})$, there exists $(v, u) \in \mathcal{N}^\vee(\pi, t_\ell) \times \mathcal{N}^\vee(\mathfrak{f}_{\pi(t_{\ell+1})})$ such that $\prod_{t \in [t_\ell, t_{\ell+1}]} u_t = vu$. To see this, one decomposes $u_{t_\ell}$ as $vu'$ according to the Zappa–Szép product and one defines $u$ as the product of $u'$ and of the $u_t$ for $t \in ]t_\ell, t_{\ell+1}]$. Assembling these pieces (and an analogous statement over
When $\mathfrak{f}$ and $\mathfrak{f}'$ are two faces of the affine Coxeter complex such that $\mathfrak{f}$ is contained in the closure $\bar{\mathfrak{f}}'$ of $\mathfrak{f}'$, we denote by $\Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')$ the set of all positive affine coroots $\beta$ such that $\mathfrak{f} \subseteq H_\beta$ and $\mathfrak{f}' \not\subseteq H_\beta^-$. We denote by $\mathcal{N}^\vee(\mathfrak{f}, \mathfrak{f}')$ the subgroup of $\mathcal{N}^\vee(\mathcal{K})$ generated by the elements $x_\beta(a)$ with $\beta \in \Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')$ and $a \in \mathbb{C}$.

We recall that a group $\Gamma$ is the (internal) Zappa–Szép product of two subgroups $\Gamma'$ and $\Gamma''$ if the product in $\Gamma$ induces a bijection $\Gamma' \times \Gamma'' \to \Gamma$. We indicate this situation with the notation $\Gamma = \Gamma' \bowtie \Gamma''$.

The following result is Proposition 19 (ii) in [2].

**Lemma 4.7.** Let $\mathfrak{f}$ and $\mathfrak{f}'$ be two faces of the affine Coxeter complex such that $\mathfrak{f} \subseteq \bar{\mathfrak{f}}'$. Then $\mathcal{N}^\vee(\mathfrak{f}) = \mathcal{N}^\vee(\mathfrak{f}, \mathfrak{f}') \bowtie \mathcal{N}^\vee(\mathfrak{f}')$, and the map

$$ \mathbb{C}^{\Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}') \to \mathcal{N}^{\vee}(\mathfrak{f}, \mathfrak{f}')}, \quad (a_\beta) \mapsto \prod_{\beta \in \Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')} x_\beta(a_\beta), $$

is bijective, whichever order on $\Phi_a^{\mathfrak{f},+}(\mathfrak{f}, \mathfrak{f}')$ is used to compute the product.

Given $\pi \in \Pi$ and $t \in [0, 1[$, we denote by $\mathfrak{f}_{\pi(t+0)}$ the face in the affine Coxeter complex that contains the points $\pi(t+h)$ for all small enough $h > 0$. Obviously, its closure meets, hence contains, the face $\mathfrak{f}_{\pi(t)}$. We set $\Phi_a^{\mathfrak{f},+}(\pi, t) = \Phi_a^{\mathfrak{f},+}(\mathfrak{f}_{\pi(t)}, \mathfrak{f}_{\pi(t+0)})$ and $\mathcal{N}^\vee(\pi, t) = \mathcal{N}^\vee(\mathfrak{f}_{\pi(t)}, \mathfrak{f}_{\pi(t+0)})$.

Concretely, $\Phi_a^{\mathfrak{f},+}(\pi, t)$ is the set of all $\beta \in \Phi_a^{\mathfrak{f},+}$ such that $\pi$ quits the half-space $H_\beta^-$ at time $t$ and $\mathcal{N}^\vee(\pi, t)$ is the subgroup of $\mathcal{N}^\vee(\mathcal{K})$ generated by the elements $x_\beta(a)$ with $\beta \in \Phi_a^{\mathfrak{f},+}(\pi, t)$ and $a \in \mathbb{C}$. Note that $\Phi_a^{\mathfrak{f},+}(\pi, t)$ is empty save for finitely many $t$.

**Proposition 4.8.** Let $(\pi_1, \dots, \pi_n) \in \Pi^n$. Then $\mathring{\mathbb{Z}}(\pi_1 \otimes \dots \otimes \pi_n)$ is the set of all elements

$$ \left[ \left( \prod_{t_1 \in [0,1[} v_{1,t_1} \right) z^{\mathrm{wt}(\pi_1)}, \dots, \left( \prod_{t_n \in [0,1[} v_{n,t_n} \right) z^{\mathrm{wt}(\pi_n)} \right] $$

with

$$ ((v_{1,t_1}, \dots, v_{n,t_n}) \in \prod_{t_1 \in [0,1[} \mathcal{N}^{\vee}(\pi_1, t_1) \times \dots \times \prod_{t_n \in [0,1[} \mathcal{N}^{\vee}(\pi_n, t_n)). $$

*Proof.* Let $\pi \in \Pi$. Let $(t_1, \dots, t_m)$ be the ordered list of all elements $t \in [0, 1[$ such that $\Phi_a^{\mathfrak{f},+}(\pi, t) \neq \emptyset$ and set $t_{m+1} = 1$.

Pick $\ell \in \{1, \dots, m\}$; between the times $t_\ell$ and $t_{\ell+1}$, the path $\pi$ never quits the half-space $H_\beta^-$ of a positive affine coroot $\beta$; as a consequence, the map $t \mapsto \mathcal{N}^\vee(\mathfrak{f}_{\pi(t)})$ is nondecreasing on $]t_\ell, t_{\ell+1}]$. It is also nondecreasing on $[0, t_1]$ if $t_1 > 0$. However, when $t$ goes past the point $t_\ell$, the group $\mathcal{N}^\vee(\mathfrak{f}_{\pi(t)})$ loses the first factor of the Zappa–Szép product $\mathcal{N}^\vee(\mathfrak{f}_{\pi(t_\ell)}) = \mathcal{N}^\vee(\pi, t_\ell) \bowtie \mathcal{N}^\vee(\mathfrak{f}_{\pi(t_\ell+0)})$.

It follows that for any family $(u_t)$ in $\prod'_{t \in [t_\ell, t_{\ell+1}]} \mathcal{N}^\vee(\mathfrak{f}_{\pi(t)})$, there exists $(v, u) \in \mathcal{N}^\vee(\pi, t_\ell) \times \mathcal{N}^\vee(\mathfrak{f}_{\pi(t_{\ell+1})})$ such that $\prod_{t \in [t_\ell, t_{\ell+1}]} u_t = vu$. To see this, one decomposes $u_{t_\ell}$ as $vu'$ according to the Zappa–Szép product and one defines $u$ as the product of $u'$ and of the $u_t$ for $t \in ]t_\ell, t_{\ell+1}]$. Assembling these pieces (and an analogous statement over
the interval $[0, t_1]$ if $t_1 > 0)$ from left to right, and noting that $N^\vee(f_{\pi(1)})$ stabilizes $L_{\mathrm{wt}(\pi)}$,
we deduce that $\mathring{\mathbb{Z}}(\pi)$ is the image of the map

$$
\prod_{t \in [0,1[} \mathcal{N}^{\vee}(\pi, t) \to \mathrm{Gr}, \quad (v_t) \mapsto \left( \prod_{t \in [0,1[} v_t \right) L_{\mathrm{wt}(\pi)}.
$$

This proves our statement in the case of just one path. The general case then follows from
Proposition 4.5 (iii).

4.4. Isomorphisms of crystals

In the previous section we explained how to build elements in $*_\mathcal{L}(\lambda)_\mu$, while in Sect. 3 we were dealing with MV cycles in $\mathcal{L}(\lambda)$. This clumsiness is due to a mismatch between the definition of the path model and the conventions in [3, 39]. To mitigate the disagreement, we define a crystal structure on

$$
*_\mathcal{L}(\lambda) = \bigsqcup_{\mu \in \Lambda} *\mathcal{L}(\lambda)_\mu.
$$

Recall the setup of Sect. 3.3: we consider a subset $J \subseteq I$, define an action of $\mathbb{C}^\times$
on Gr given by a special dominant weight $\theta_J$, and get the diagram (4). Now let $(\lambda, \nu) \in
(\Lambda^+)^n \times \Lambda$, let $\zeta$ be the coset of $\nu$ modulo $\mathbb{Z}\Phi_J$, and let $Z \in *\mathcal{L}(\lambda)_\nu$. Then $m_n(Z) \subseteq$
$S_\nu \subseteq \mathrm{Gr}^+_{J,\zeta}$ and there is a unique weight $\mu \in \Lambda_J^+$ characterized by the conditions

$$
(p_{J,\zeta} \circ m_n)(Z) \subseteq \overline{\mathrm{Gr}}_{J}^{\mu} \quad \text{and} \quad (p_{J,\zeta} \circ m_n)(Z) \cap \mathrm{Gr}_{J}^{\mu} \neq \emptyset.
$$

We denote this weight by $\mu_J(Z)$.

By analogy with Proposition 3.7, we can then claim the existence of a crystal structure
on $*_\mathcal{L}(\lambda)$ such that for all $\nu \in \Lambda$, $i \in I$ and $Z \in \mathcal{L}(\lambda)_\nu$,

• we have wt(Z) = v, εᵢ(Z) = 1/2 ⟨αᵢrait, μ{ᵢ}(Z) - v⟩ and φᵢ(Z) = 1/2 ⟨αᵢrait, μ{ᵢ}(Z) + v⟩;

• if $Y \in *\mathcal{L}(\lambda)_{v+\alpha_i}$, then $Y = \tilde{e}_i Z$ if and only if $\bar{Y} \supseteq Z$ and $\mu_{\{i\}}(Y) = \mu_{\{i\}}(Z)$.

**Theorem 4.9.** Let $(\lambda_1, \dots, \lambda_n) \in (\Lambda^+)^n$, and for each $j \in \{1, \dots, n\}$ choose a subcrys-
tal $\Pi_j$ of $\Pi$ isomorphic to $B(\lambda_j)$. Then the map $(\pi_1, \dots, \pi_n) \mapsto \mathbb{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ is an
isomorphism of crystals

$$
\Pi_1 \otimes \cdots \otimes \Pi_n \xrightarrow{\approx} *\mathcal{L}(\lambda_1, \dots, \lambda_n).
$$

Proof. Let $i \in I$ and let $(\pi_1, \dots, \pi_n) \in \Pi_1 \times \dots \times \Pi_n$. Set $v = \mathrm{wt}(\pi_1) + \dots + \mathrm{wt}(\pi_n)$, let $\zeta$ be the coset of $v$ modulo $\mathbb{Z}\alpha_i$, and set $\pi = \pi_1 * \dots * \pi_n$,

$$
p = \min \{ \langle \alpha_i^{\vee}, \pi(t) \rangle \mid t \in [0, 1] \} \quad \text{and} \quad q = \langle \alpha_i^{\vee}, v \rangle = \langle \alpha_i^{\vee}, \pi(1) \rangle.
$$

For any $a \in \mathbb{C}[z, z^{-1}]$ and any positive coroot $\alpha^{\vee}$, we have

$$
\lim_{c \to 0} \theta_{\{i\}}(c) x_{\alpha^{\vee}}(a) \theta_{\{i\}}(c)^{-1} = \begin{cases} x_{\alpha^{\vee}}(a) & \text{if } \alpha^{\vee} = \alpha_i^{\vee}, \\ 1 & \text{otherwise.} \end{cases}
$$
the interval $[0, t_1]$ if $t_1 > 0)$ from left to right, and noting that $N^\vee(f_{\pi(1)})$ stabilizes $L_{\mathrm{wt}(\pi)}$,
we deduce that $\mathring{\mathbb{Z}}(\pi)$ is the image of the map

$$
\prod_{t \in [0,1[} \mathcal{N}^{\vee}(\pi, t) \to \mathrm{Gr}, \quad (v_t) \mapsto \left( \prod_{t \in [0,1[} v_t \right) L_{\mathrm{wt}(\pi)}.
$$

This proves our statement in the case of just one path. The general case then follows from
Proposition 4.5 (iii).

4.4. Isomorphisms of crystals

In the previous section we explained how to build elements in $*_\mathcal{L}(\lambda)_\mu$, while in Sect. 3 we were dealing with MV cycles in $\mathcal{L}(\lambda)$. This clumsiness is due to a mismatch between the definition of the path model and the conventions in [3, 39]. To mitigate the disagreement, we define a crystal structure on

$$
*_\mathcal{L}(\lambda) = \bigsqcup_{\mu \in \Lambda} *\mathcal{L}(\lambda)_\mu.
$$

Recall the setup of Sect. 3.3: we consider a subset $J \subseteq I$, define an action of $\mathbb{C}^\times$
on Gr given by a special dominant weight $\theta_J$, and get the diagram (4). Now let $(\lambda, v) \in$
$(\Lambda^+)^n \times \Lambda$, let $\zeta$ be the coset of $v$ modulo $\mathbb{Z}\Phi_J$, and let $Z \in *\mathcal{L}(\lambda)_v$. Then $m_n(Z) \subseteq$
$S_v \subseteq \mathrm{Gr}^{+}_{J,\zeta}$ and there is a unique weight $\mu \in \Lambda_J^+$ characterized by the conditions

$$
(p_{J,\zeta} \circ m_n)(Z) \subseteq \overline{\mathrm{Gr}}_{J}^{\mu} \quad \text{and} \quad (p_{J,\zeta} \circ m_n)(Z) \cap \mathrm{Gr}_{J}^{\mu} \neq \emptyset.
$$

We denote this weight by $\mu_J(Z)$.

By analogy with Proposition 3.7, we can then claim the existence of a crystal structure
on $*_\mathcal{L}(\lambda)$ such that for all $\nu \in \Lambda$, $i \in I$ and $Z \in \mathcal{L}(\lambda)_\nu$,

• we have wt(Z) = v, εᵢ(Z) = 1/2 ⟨αᵢrait, μ{ᵢ}(Z) - v⟩ and φᵢ(Z) = 1/2 ⟨αᵢrait, μ{ᵢ}(Z) + v⟩;

• if $Y \in *\mathcal{L}(\lambda)_{v+\alpha_i}$, then $Y = \tilde{e}_i Z$ if and only if $\bar{Y} \supseteq Z$ and $\mu_{\{i\}}(Y) = \mu_{\{i\}}(Z)$.

**Theorem 4.9.** Let $(\lambda_1, \dots, \lambda_n) \in (\Lambda^+)^n$, and for each $j \in \{1, \dots, n\}$ choose a subcrys-
tal $\Pi_j$ of $\Pi$ isomorphic to $B(\lambda_j)$. Then the map $(\pi_1, \dots, \pi_n) \mapsto \mathbb{Z}(\pi_1 \otimes \dots \otimes \pi_n)$ is an
isomorphism of crystals

$$
\Pi_1 \otimes \cdots \otimes \Pi_n \xrightarrow{\approx} *\mathcal{L}(\lambda_1, \dots, \lambda_n).
$$

Proof. Let $i \in I$ and let $(\pi_1, \dots, \pi_n) \in \Pi_1 \times \dots \times \Pi_n$. Set $v = \mathrm{wt}(\pi_1) + \dots + \mathrm{wt}(\pi_n)$, let $\zeta$ be the coset of $v$ modulo $\mathbb{Z}\alpha_i$, and set $\pi = \pi_1 * \dots * \pi_n$,

$$
p = \min \{ \langle \alpha_i^{\vee}, \pi(t) \rangle \mid t \in [0, 1] \} \quad \text{and} \quad q = \langle \alpha_i^{\vee}, v \rangle = \langle \alpha_i^{\vee}, \pi(1) \rangle.
$$

For any $a \in \mathbb{C}[z, z^{-1}]$ and any positive coroot $\alpha^\vee$, we have

$$
\lim_{c \to 0} \theta_{\{i\}}(c) x_{\alpha^c}(a) \theta_{\{i\}}(c)^{-1} = \begin{cases} x_{\alpha^c}(a) & \text{if } \alpha^c = \alpha_i^c, \\ 1 & \text{otherwise.} \end{cases}
$$
Using Proposition 4.8, we see that $p_{\{i\},\zeta}(\mathring{\mathcal{Z}}(\pi))$ is the set of all elements of the form

$$ \lim_{c \to 0} \prod_{t \in [0,1[} \left( \prod_{\beta \in \Phi_a^{\vee} + (\pi, t)} \theta_{\{i\}}(c) x_{\beta}(a_{t,\beta}) \theta_{\{i\}}(c)^{-1} \right) L_v $$

where $a_{t,\beta}$ are complex numbers. All factors in the product disappear in the limit $c \to 0$, except those for the affine roots $\beta$ of direction $\alpha_i^{\vee}$. Let $(\alpha_i^{\vee}, p_1), \dots, (\alpha_i^{\vee}, p_s)$ be these affine roots. Since the function $t \mapsto \langle \alpha_i^{\vee}, \pi(t) \rangle$ assumes the value $p$ and therefore reaches the value $q$, the path $\pi$ must, at some point, quit each half-space $H_{(\alpha_i^{\vee}, p)}^{-}$, $H_{(\alpha_i^{\vee}, p+1)}^{-}, \dots, H_{(\alpha_i^{\vee}, q-1)}^{-}$, so

$$ \{p, p+1, \dots, q-1\} \subseteq \{p_1, \dots, p_s\} \subseteq \{p, p+1, \dots\}. $$

We conclude that

$$ p_{\{i\},\zeta}(\mathring{\mathcal{Z}}(\pi)) = \{x_{\alpha_i^{\vee}}(az^p)L_v \mid a \in \mathcal{O}/z^{q-p}\mathcal{O}\}. $$

Proposition 4.5 (iii) and a variant of Proposition 3.6 then jointly imply

$$ (p_{\{i\},\zeta} \circ m_n)(\mathring{\mathcal{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)) = p_{\{i\},\zeta}(\mathring{\mathcal{Z}}(\pi)) = \overline{\mathrm{Gr}_{\{i\}}^{\mu}} \cap S_{\{i\},v} $$

where $\mu = v - p\alpha_i$. Thus,

$$ \mu_{\{i\}}(\mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)) = v - p\alpha_i \quad (15) $$

and

$$ \varepsilon_i(\mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)) = -p \quad \text{and} \quad \varphi_i(\mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)) = q - p. $$

These equations show that the map $(\pi_1, \dots, \pi_n) \mapsto \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ is compatible with the functions $\varepsilon_i$ and $\varphi_i$.

Now compute

$$ \eta_1 \otimes \cdots \otimes \eta_n = \tilde{e}_i(\pi_1 \otimes \cdots \otimes \pi_n) $$

in the crystal $\Pi_1 \otimes \cdots \otimes \Pi_n$, assuming this is doable. By Proposition 4.5 (iv),

$$ \overline{\mathfrak{Z}(\eta_1 \otimes \cdots \otimes \eta_n)} \supseteq \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n). \quad (16) $$

Let $\eta = \eta_1 * \cdots * \eta_n$. Then $\eta = \tilde{e}_i\pi$ by Proposition 4.2 (ii), and therefore

$$ \mathrm{wt}(\eta) = v + \alpha_i \quad \text{and} \quad \min \{\langle \alpha_i^{\vee}, \eta(t) \rangle \mid t \in [0,1]\} = p+1. $$

Repeating the arguments above, we get

$$ \mu_{\{i\}}(\mathfrak{Z}(\eta_1 \otimes \cdots \otimes \eta_n)) = (v + \alpha_i) - (p+1)\alpha_i = v - p\alpha_i. $$

Together with (15) and (16), this gives

$$ \mathfrak{Z}(\eta_1 \otimes \cdots \otimes \eta_n) = \tilde{e}_i \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n). $$

We conclude that the map $(\pi_1, \dots, \pi_n) \mapsto \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ has the required compatibility with the operations $\tilde{e}_i$. ■
Using Proposition 4.8, we see that $p_{\{i\},\xi}(\mathring{\mathcal{Z}}(\pi))$ is the set of all elements of the form

$$ \lim_{c \to 0} \prod_{t \in [0,1[} \left( \prod_{\beta \in \Phi_a^{\vee} + (\pi, t)} \theta_{\{i\}}(c) x_{\beta}(a_{t,\beta}) \theta_{\{i\}}(c)^{-1} \right) L_v $$

where $a_{t,\beta}$ are complex numbers. All factors in the product disappear in the limit $c \to 0$, except those for the affine roots $\beta$ of direction $\alpha_i^{\vee}$. Let $(\alpha_i^{\vee}, p_1), \dots, (\alpha_i^{\vee}, p_s)$ be these affine roots. Since the function $t \mapsto \langle \alpha_i^{\vee}, \pi(t) \rangle$ assumes the value $p$ and therefore reaches the value $q$, the path $\pi$ must, at some point, quit each half-space $H_{(\alpha_i^{\vee}, p)}^{-}$, $H_{(\alpha_i^{\vee}, p+1)}^{-}, \dots, H_{(\alpha_i^{\vee}, q-1)}^{-}$, so

$$ \{p, p+1, \dots, q-1\} \subseteq \{p_1, \dots, p_s\} \subseteq \{p, p+1, \dots\}. $$

We conclude that

$$ p_{\{i\},\xi}(\mathring{\mathcal{Z}}(\pi)) = \{x_{\alpha_i^{\vee}}(az^p)L_v \mid a \in \mathcal{O}/z^{q-p}\mathcal{O}\}. $$

Proposition 4.5 (iii) and a variant of Proposition 3.6 then jointly imply

$$ (p_{\{i\},\xi} \circ m_n)(\mathring{\mathcal{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)) = p_{\{i\},\xi}(\mathring{\mathcal{Z}}(\pi)) = \overline{\mathrm{Gr}_{\{i\}}^{\mu}} \cap S_{\{i\},v} $$

where $\mu = v - p\alpha_i$. Thus,

$$ \mu_{\{i\}}(\mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)) = v - p\alpha_i \quad (15) $$

and

$$ \varepsilon_i(\mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)) = -p \quad \text{and} \quad \varphi_i(\mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)) = q - p. $$

These equations show that the map $(\pi_1, \dots, \pi_n) \mapsto \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ is compatible with the functions $\varepsilon_i$ and $\varphi_i$.

Now compute

$$ \eta_1 \otimes \cdots \otimes \eta_n = \tilde{e}_i(\pi_1 \otimes \cdots \otimes \pi_n) $$

in the crystal $\Pi_1 \otimes \cdots \otimes \Pi_n$, assuming this is doable. By Proposition 4.5 (iv),

$$ \overline{\mathfrak{Z}(\eta_1 \otimes \cdots \otimes \eta_n)} \supseteq \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n). \quad (16) $$

Let $\eta = \eta_1 * \cdots * \eta_n$. Then $\eta = \tilde{e}_i\pi$ by Proposition 4.2 (ii), and therefore

$$ \mathrm{wt}(\eta) = v + \alpha_i \quad \text{and} \quad \min \{\langle \alpha_i^{\vee}, \eta(t) \rangle \mid t \in [0, 1]\} = p+1. $$

Repeating the arguments above, we get

$$ \mu_{\{i\}}(\mathfrak{Z}(\eta_1 \otimes \cdots \otimes \eta_n)) = (v + \alpha_i) - (p+1)\alpha_i = v - p\alpha_i. $$

Together with (15) and (16), this gives

$$ \mathfrak{Z}(\eta_1 \otimes \cdots \otimes \eta_n) = \tilde{e}_i \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n). $$

We conclude that the map $(\pi_1, \dots, \pi_n) \mapsto \mathfrak{Z}(\pi_1 \otimes \cdots \otimes \pi_n)$ has the required compatibility with the operations $\tilde{e}_i$. ■
**Corollary 4.10.** Let $(\lambda_1, \dots, \lambda_n) \in (\Lambda^+)^n$. Then the map $(Z_1, \dots, Z_n) \mapsto \overline{\Psi(Z_1 \times \cdots \times Z_n)}$ from Proposition 2.2 (ii) is an isomorphism of crystals

$$ *\mathcal{L}(\lambda_1) \otimes \cdots \otimes *\mathcal{L}(\lambda_n) \xrightarrow{\approx} *\mathcal{L}(\lambda_1, \dots, \lambda_n). $$

The crystals $\mathcal{L}(\lambda)$ enjoy a factorization property analogous to Corollary 4.10; one must however use the opposite tensor product on crystals.

**Remark 4.11.** The plactic algebra [32] is an algebraic combinatorial tool invented by Lascoux and Schützenberger long before the notion of a crystal basis of a representation was introduced. Loosely speaking, for a complex reductive algebraic group, the plactic algebra is the algebra having as basis the union $\cup_{\lambda \in \Lambda^+} B(\lambda)$ of the crystal bases $B(\lambda)$ for all irreducible representations, the product being given by the tensor product of crystals. For $G = \text{SL}_n(\mathbb{C})$, Lascoux and Schützenberger give a description of such an algebra in terms of the word algebra modulo the Knuth relations, and it was shown later that this algebra is isomorphic to the one given by the crystal basis. A combinatorial Lascoux-Schützenberger type description for the other types was given in [34]; this description uses the path model.

It is natural to ask whether it is possible to do the same with MV cycles: endow the set of all MV cycles for all dominant weights $\lambda \in \Lambda^+$ with the structure of a crystal and define (in a geometric way) a multiplication on the cycles which mimics the plactic algebra. For $G = \text{SL}_n(\mathbb{C})$, a positive answer was given in [18]. This approach was adapted to the symplectic case in [44].

The results in this section can be naturally viewed as a generalization of [18] to arbitrary connected reductive groups. Using [34] and Proposition 4.2, one can use the set $\Pi$ to construct the plactic algebra so that it has as basis equivalence classes (generalized Knuth relations) of elements in $\Pi$. The sets $\mathring{\mathcal{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ (Proposition 4.5) replace in the general setting the Bialynicki-Birula cells in [18]. By combining Proposition 4.5 (iii) and Theorem 4.9, we see that the closure of $m_n(\mathring{\mathcal{Z}}(\pi_1 \otimes \cdots \otimes \pi_n))$ is an MV cycle which depends only on the class of the path $\pi_1 * \cdots * \pi_n$ modulo the generalized Knuth relations. In particular, the main result of [18] follows as a special case.

A different approach to this problem was taken by Xiao and Zhu [45]. They define a set of ‘elementary Littelmann paths’, modeled over minuscule or quasi-minuscule representations, use the methods from [40] to assign an MV cycle to each concatenation of elementary Littelmann paths, and show that the resulting map factorizes through the generalized Knuth relations.

## 4.5. *Proof of Proposition 4.5 (iv)*

This section can be skipped without substantial loss of appreciation of our main storyline. We follow the same method as in [2, proof of Proposition 5.11].

The group $G^\vee(\mathbb{C})$ is generated by elements $x_{\alpha^\vee}(a)$ and $c^\lambda$, where $(a, \alpha^\vee) \in \mathbb{C} \times \Phi^\vee$ and $(c, \lambda) \in \mathbb{C}^\times \times \Lambda$, which obey the following relations:
**Corollary 4.10.** Let $(\lambda_1, \dots, \lambda_n) \in (\Lambda^+)^n$. Then the map $(Z_1, \dots, Z_n) \mapsto \overline{\Psi(Z_1 \times \cdots \times Z_n)}$ from Proposition 2.2 (ii) is an isomorphism of crystals

$$ \ast \mathcal{L}(\lambda_1) \otimes \cdots \otimes \ast \mathcal{L}(\lambda_n) \xrightarrow{\approx} \ast \mathcal{L}(\lambda_1, \dots, \lambda_n). $$

The crystals $\mathcal{L}(\lambda)$ enjoy a factorization property analogous to Corollary 4.10; one must however use the opposite tensor product on crystals.

**Remark 4.11.** The plactic algebra [32] is an algebraic combinatorial tool invented by Lascoux and Schützenberger long before the notion of a crystal basis of a representation was introduced. Loosely speaking, for a complex reductive algebraic group, the plactic algebra is the algebra having as basis the union $\cup_{\lambda \in \Lambda^+} B(\lambda)$ of the crystal bases $B(\lambda)$ for all irreducible representations, the product being given by the tensor product of crystals. For $G = \text{SL}_n(\mathbb{C})$, Lascoux and Schützenberger give a description of such an algebra in terms of the word algebra modulo the Knuth relations, and it was shown later that this algebra is isomorphic to the one given by the crystal basis. A combinatorial Lascoux-Schützenberger type description for the other types was given in [34]; this description uses the path model.

It is natural to ask whether it is possible to do the same with MV cycles: endow the set of all MV cycles for all dominant weights $\lambda \in \Lambda^+$ with the structure of a crystal and define (in a geometric way) a multiplication on the cycles which mimics the plactic algebra. For $G = \text{SL}_n(\mathbb{C})$, a positive answer was given in [18]. This approach was adapted to the symplectic case in [44].

The results in this section can be naturally viewed as a generalization of [18] to arbitrary connected reductive groups. Using [34] and Proposition 4.2, one can use the set $\Pi$ to construct the plactic algebra so that it has as basis equivalence classes (generalized Knuth relations) of elements in $\Pi$. The sets $\mathring{\mathcal{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ (Proposition 4.5) replace in the general setting the Bialynicki-Birula cells in [18]. By combining Proposition 4.5 (iii) and Theorem 4.9, we see that the closure of $m_n(\mathring{\mathcal{Z}}(\pi_1 \otimes \cdots \otimes \pi_n))$ is an MV cycle which depends only on the class of the path $\pi_1 * \cdots * \pi_n$ modulo the generalized Knuth relations. In particular, the main result of [18] follows as a special case.

A different approach to this problem was taken by Xiao and Zhu [45]. They define a set of ‘elementary Littelmann paths’, modeled over minuscule or quasi-minuscule representations, use the methods from [40] to assign an MV cycle to each concatenation of elementary Littelmann paths, and show that the resulting map factorizes through the generalized Knuth relations.

## 4.5. *Proof of Proposition 4.5 (iv)*

This section can be skipped without substantial loss of appreciation of our main storyline. We follow the same method as in [2, proof of Proposition 5.11].

The group $G^\vee(\mathbb{C})$ is generated by elements $x_{\alpha^\vee}(a)$ and $c^\lambda$, where $(a, \alpha^\vee) \in \mathbb{C} \times \Phi^\vee$ and $(c, \lambda) \in \mathbb{C}^\times \times \Lambda$, which obey the following relations:
• For any $(a, \alpha^{\vee}) \in \mathbb{C} \times \Phi^{\vee}$ and any $(c, \lambda) \in \mathbb{C}^{\times} \times \Lambda$,

$$c^{\lambda}_{x_{\alpha^{\vee}}(a)}c^{-\lambda} = x_{\alpha^{\vee}}(c^{\langle \alpha^{\vee}, \lambda \rangle}a).$$

• Given two linearly independent elements $\alpha^{\vee}$ and $\beta^{\vee}$ in $\Phi^{\vee}$, there exist constants $C_{i,j}$ such that

$$x_{\alpha^{\vee}}(a)x_{\beta^{\vee}}(b)x_{\alpha^{\vee}}(a)^{-1}x_{\beta^{\vee}}(b)^{-1} = \prod_{(i,j)} x_{i\alpha^{\vee}+j\beta^{\vee}}(C_{i,j}a^ib^j) \quad (17)$$

for any $(a, b) \in \mathbb{C}^2$. The product on the right-hand side is taken over all pairs of positive integers $(i, j)$ for which $i\alpha^{\vee} + j\beta^{\vee} \in \Phi^{\vee}$, in the order of increasing $i + j$.

Further, the one-parameter subgroups $x_{\alpha^{\vee}}$ can be normalized so that for any root $\alpha \in \Phi$,

• for any $(a, b) \in \mathbb{C}^2$ such that $1 - ab \neq 0$,

$$x_{\alpha^{\vee}}(a)x_{-\alpha^{\vee}}(b) = x_{-\alpha^{\vee}}(b/(1-ab))(1-ab)^{\alpha}x_{\alpha^{\vee}}(a/(1-ab)); \quad (18)$$

• there exists an element $\overline{s_{\alpha}} \in G^{\vee}(\mathbb{C})$ such that for any $a \in \mathbb{C}^{\times}$,

$$x_{\alpha^{\vee}}(a)x_{-\alpha^{\vee}}(a^{-1})x_{\alpha^{\vee}}(a) = x_{-\alpha^{\vee}}(a^{-1})x_{\alpha^{\vee}}(a)x_{-\alpha^{\vee}}(a^{-1}) = a^{\alpha} \overline{s_{\alpha}} = \overline{s_{\alpha}} a^{-\alpha}. \quad (19)$$

This element $\overline{s_{\alpha}}$ lifts to the normalizer of $T^{\vee}(\mathbb{C})$ the reflection $s_{\alpha} \in W$ along the root $\alpha$. All the above relations also hold for scalars $b, c$ in $\mathcal{K}$, provided of course that we regard them in $G^{\vee}(\mathcal{K})$.

The Chevalley commutation relation (17) implies the following easy lemma.

**Lemma 4.12.** Let $\mathfrak{f}$ be a face of the affine Coxeter complex, let $(\alpha^{\vee}, p)$ and $(\beta^{\vee}, q)$ be two positive affine coroots, and let $(a, b) \in \mathcal{O}^2$. Assume that $\alpha^{\vee}$ is simple, $\alpha^{\vee} \neq \beta^{\vee}$, and

$$\mathfrak{f} \subseteq \overline{H}_{(-\alpha^{\vee}, -p)} \cap \overline{H}_{(\beta^{\vee}, q)}.$$

Then

$$x_{-\alpha^{\vee}}(az^{-p})x_{\beta^{\vee}}(bz^{q})x_{-\alpha^{\vee}}(-az^{-p}) \in N^{\vee}(\mathfrak{f}).$$

*Proof.* We consider the situation set forth in the statement of the lemma. Using (17), we write

$$x_{-\alpha^{\vee}}(az^{-p})x_{\beta^{\vee}}(bz^{q})x_{-\alpha^{\vee}}(-az^{-p})x_{\beta^{\vee}}(-bz^{q}) = \prod_{(i,j)} x_{-i\alpha^{\vee}+j\beta^{\vee}}(C_{i,j}a^ib^jz^{-ip+jq}) \quad (20)$$

where the product on the right-hand side is taken over all pairs of positive integers $(i, j)$ for which $-i\alpha^{\vee} + j\beta^{\vee}$ is a coroot.

Consider such a pair $(i, j)$. In view of our assumptions, the coroot $-i\alpha^{\vee} + j\beta^{\vee}$ is necessarily positive. Moreover, for any $x \in \mathfrak{f}$ we have

$$\langle -i\alpha^{\vee} + j\beta^{\vee}, x \rangle = i \langle -\alpha^{\vee}, x \rangle + j \langle \beta^{\vee}, x \rangle \leq i(-p) + jq,$$

so $\mathfrak{f} \subseteq \overline{H}_{(-i\alpha^{\vee}+j\beta^{\vee}, -ip+jq)}$. It follows that the right-hand side of (20) lies in $\mathcal{N}^{\vee}(\mathfrak{f})$, which readily implies the statement. ■
* For any $(a, \alpha^{\vee}) \in \mathbb{C} \times \Phi^{\vee}$ and any $(c, \lambda) \in \mathbb{C}^{\times} \times \Lambda$,

$$c^{\lambda}_{x_{\alpha^{\vee}}(a)}c^{-\lambda} = x_{\alpha^{\vee}}(c^{\langle \alpha^{\vee}, \lambda \rangle}a).$$

* Given two linearly independent elements $\alpha^{\vee}$ and $\beta^{\vee}$ in $\Phi^{\vee}$, there exist constants $C_{i,j}$ such that

$$x_{\alpha^{\vee}}(a)x_{\beta^{\vee}}(b)x_{\alpha^{\vee}}(a)^{-1}x_{\beta^{\vee}}(b)^{-1} = \prod_{(i,j)} x_{i\alpha^{\vee}+j\beta^{\vee}}(C_{i,j}a^ib^j) \quad (17)$$

for any $(a, b) \in \mathbb{C}^2$. The product on the right-hand side is taken over all pairs of positive integers $(i, j)$ for which $i\alpha^{\vee} + j\beta^{\vee} \in \Phi^{\vee}$, in the order of increasing $i + j$.

Further, the one-parameter subgroups $x_{\alpha^{\vee}}$ can be normalized so that for any root $\alpha \in \Phi$,

* for any $(a, b) \in \mathbb{C}^2$ such that $1 - ab \neq 0$,

$$x_{\alpha^{\vee}}(a)x_{-\alpha^{\vee}}(b) = x_{-\alpha^{\vee}}(b/(1-ab))(1-ab)^{\alpha}x_{\alpha^{\vee}}(a/(1-ab)); \quad (18)$$

* there exists an element $\overline{s_{\alpha}} \in G^{\vee}(\mathbb{C})$ such that for any $a \in \mathbb{C}^{\times}$,

$$x_{\alpha^{\vee}}(a)x_{-\alpha^{\vee}}(a^{-1})x_{\alpha^{\vee}}(a) = x_{-\alpha^{\vee}}(a^{-1})x_{\alpha^{\vee}}(a)x_{-\alpha^{\vee}}(a^{-1}) = a^{\alpha} \overline{s_{\alpha}} = \overline{s_{\alpha}} a^{-\alpha}. \quad (19)$$

This element $\overline{s_{\alpha}}$ lifts to the normalizer of $T^{\vee}(\mathbb{C})$ the reflection $s_{\alpha} \in W$ along the root $\alpha$. All the above relations also hold for scalars $b, c$ in $\mathcal{K}$, provided of course that we regard them in $G^{\vee}(\mathcal{K})$.

The Chevalley commutation relation (17) implies the following easy lemma.

**Lemma 4.12.** Let $\mathfrak{f}$ be a face of the affine Coxeter complex, let $(\alpha^{\vee}, p)$ and $(\beta^{\vee}, q)$ be two positive affine coroots, and let $(a, b) \in \mathcal{O}^2$. Assume that $\alpha^{\vee}$ is simple, $\alpha^{\vee} \neq \beta^{\vee}$, and

$$\mathfrak{f} \subseteq \overline{H}_{(-\alpha^{\vee}, -p)} \cap \overline{H}_{(\beta^{\vee}, q)}.$$

Then

$$x_{-\alpha^{\vee}}(az^{-p})x_{\beta^{\vee}}(bz^{q})x_{-\alpha^{\vee}}(-az^{-p}) \in N^{\vee}(\mathfrak{f}).$$

*Proof.* We consider the situation set forth in the statement of the lemma. Using (17), we write

$$x_{-\alpha^{\vee}}(az^{-p})x_{\beta^{\vee}}(bz^{q})x_{-\alpha^{\vee}}(-az^{-p})x_{\beta^{\vee}}(-bz^{q}) = \prod_{(i,j)} x_{-i\alpha^{\vee}+j\beta^{\vee}}(C_{i,j}a^ib^jz^{-ip+jq}) \quad (20)$$

where the product on the right-hand side is taken over all pairs of positive integers $(i, j)$ for which $-i\alpha^{\vee} + j\beta^{\vee}$ is a coroot.

Consider such a pair $(i, j)$. In view of our assumptions, the coroot $-i\alpha^{\vee} + j\beta^{\vee}$ is necessarily positive. Moreover, for any $x \in \mathfrak{f}$ we have

$$\langle -i\alpha^{\vee} + j\beta^{\vee}, x \rangle = i \langle -\alpha^{\vee}, x \rangle + j \langle \beta^{\vee}, x \rangle \leq i(-p) + jq,$$

so $\mathfrak{f} \subseteq \overline{H}_{(-i\alpha^{\vee}+j\beta^{\vee}, -ip+jq)}$. It follows that the right-hand side of (20) lies in $\mathcal{N}^{\vee}(\mathfrak{f})$, which readily implies the statement. ■