# Demazure Descent and Representations of Reductive Groups

Authors: Sergey Arkhipov; Tina Kanstrup
Journal: Algebra
 (2014)

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## Abstract

We introduce the notion of Demazure descent data on a triangulated categoryC and define the descent category for such data. We illustrate the definition by our basic example. Let G be a reductive algebraic group with a Borel subgroup B. Demazure functors form Demazure descent data on DbRepB and the descent category is equivalent to DbRepG.


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## Body

## 1. Motivation

The present paper is the first one in a series devoted to various cases of categorical descent. Philosophically, our interest in the subject grew out of attempts to understand the main construction from the recent paper by Ben-Zvi and Nadler [1] in plain terms that would not involve higher category theory.


### 1.1. Beilinson-Bernstein Localization and Derived Descent

LetG be a reductive algebraic group with the Lie algebra g. Denote the Flag variety of G by Fl. A major part of Geometric Representation Theory originated in the seminal work of Beĭlinson and Bernstein [2] devoted to investigation of the globalization functor D-mod(Fl)→U(g)-mod. This functor turns out to be fully faithful and provides geometric and topological tools to investigate a wide class of U(g)-modules, in particular the ones from the famous category O. Various generalizations of this result lead to the investigation of the categories of twisted D-modules on the Flag variety and on the base affine space for G and of their derived categories.Ben-Zvi and Nadler define a certain comonad acting on a higher categorical version for the derived category of D-modules on the base affine space. In fact, the functor is built into the higher categorical treatment of Beilinson-Bernstein localization-globalization construction.Using the heavy machinery of Barr-Beck-Lurie descent, the authors argue that the derived category ofU(g)-modules is equivalent to the category of D-modules equivariant with respect to this comonad. Thus the global sections functor becomes equivariant with respect to the action. The comonad is called the Hecke comonad. It provides a categorification for the classical action of the Weyl group on various homological and K-theoretic invariants of the Flag variety.Notice that the descent construction fails to work on the level of the usual triangulated categories. Ideally one would like to replace it by a categorical action of the Weyl group or rather of the Braid group on categories of D-modules related to the Flag variety. One would need to define a notion of “invariants” with respect to such action.


### 1.2. Descent in EquivariantK-Theory

Another source of inspiration for the present paper, which is in a way closer to our work, is a recent article of Harada et al. [3]. Given a compact space X with an action of a compact reductive Lie group G, the authors express the G-equivariant K-theory of X via the T-equivariant one. Here T denotes a fixed maximal torus in G. Harada et al. show that the natural action of the Weyl group W on KT(X) extends to an action of a degenerate Hecke ring generated by divided difference operators which was introduced earlier in the context of Schubert calculus by Demazure. The operators are called Demazure operators.The main result in the paper [3] states that the ring KG(X) is isomorphic to the subring of KT(X) annihilated by the augmentation ideal in the degenerate Hecke algebra. In other words, a T-equivariant class is G-equivariant if and only if it is killed by the Demazure operators.In the present paper, we define a notion of Demazure descent on a triangulated categoryC. Thus Demazure operators are replaced by Demazure functors. These functors satisfy a categorified version of degenerate Hecke algebra relations and form a Demazure descent data on C. We define the descent category for such data. Demazure descent is supposed to be a technique replacing the naive notion of Weyl group invariants, on the categorical level.We provide the first example of Demazure descent. Consider a reductive algebraic groupG and fix a Borel subgroup B⊂G. Categorifying the construction form [3], we consider Demazure functors Dsi acting on the derived category of B-modules. We prove that the functors form a Demazure descent data and identify the descent category with the derived category of G-modules.


## 1.1. Beilinson-Bernstein Localization and Derived Descent

LetG be a reductive algebraic group with the Lie algebra g. Denote the Flag variety of G by Fl. A major part of Geometric Representation Theory originated in the seminal work of Beĭlinson and Bernstein [2] devoted to investigation of the globalization functor D-mod(Fl)→U(g)-mod. This functor turns out to be fully faithful and provides geometric and topological tools to investigate a wide class of U(g)-modules, in particular the ones from the famous category O. Various generalizations of this result lead to the investigation of the categories of twisted D-modules on the Flag variety and on the base affine space for G and of their derived categories.Ben-Zvi and Nadler define a certain comonad acting on a higher categorical version for the derived category of D-modules on the base affine space. In fact, the functor is built into the higher categorical treatment of Beilinson-Bernstein localization-globalization construction.Using the heavy machinery of Barr-Beck-Lurie descent, the authors argue that the derived category ofU(g)-modules is equivalent to the category of D-modules equivariant with respect to this comonad. Thus the global sections functor becomes equivariant with respect to the action. The comonad is called the Hecke comonad. It provides a categorification for the classical action of the Weyl group on various homological and K-theoretic invariants of the Flag variety.Notice that the descent construction fails to work on the level of the usual triangulated categories. Ideally one would like to replace it by a categorical action of the Weyl group or rather of the Braid group on categories of D-modules related to the Flag variety. One would need to define a notion of “invariants” with respect to such action.


## 1.2. Descent in EquivariantK-Theory

Another source of inspiration for the present paper, which is in a way closer to our work, is a recent article of Harada et al. [3]. Given a compact space X with an action of a compact reductive Lie group G, the authors express the G-equivariant K-theory of X via the T-equivariant one. Here T denotes a fixed maximal torus in G. Harada et al. show that the natural action of the Weyl group W on KT(X) extends to an action of a degenerate Hecke ring generated by divided difference operators which was introduced earlier in the context of Schubert calculus by Demazure. The operators are called Demazure operators.The main result in the paper [3] states that the ring KG(X) is isomorphic to the subring of KT(X) annihilated by the augmentation ideal in the degenerate Hecke algebra. In other words, a T-equivariant class is G-equivariant if and only if it is killed by the Demazure operators.In the present paper, we define a notion of Demazure descent on a triangulated categoryC. Thus Demazure operators are replaced by Demazure functors. These functors satisfy a categorified version of degenerate Hecke algebra relations and form a Demazure descent data on C. We define the descent category for such data. Demazure descent is supposed to be a technique replacing the naive notion of Weyl group invariants, on the categorical level.We provide the first example of Demazure descent. Consider a reductive algebraic groupG and fix a Borel subgroup B⊂G. Categorifying the construction form [3], we consider Demazure functors Dsi acting on the derived category of B-modules. We prove that the functors form a Demazure descent data and identify the descent category with the derived category of G-modules.


## 2. The Setting

### 2.1. Root Data

LetG be a reductive algebraic group over an algebraically closed field k of characteristic zero. Let T be a Cartan subgroup of G and let (I,X,Y) be the corresponding root data, where I is the set of vertices of the Dynkin diagram, X is the weight lattice of G, and Y is the coroot lattice of G. Choose a Borel subgroup T⊂B⊂G. Denote the set of roots for G by Φ=Φ+⊔Φ-. Let {α1,…,αn} be the set of simple roots. The Weyl group W=Norm(T)/T of the fixed maximal torus acts naturally on the lattices X and Y and on the R-vector spaces spanned by them, by reflections in root hyperplanes. The simple reflection corresponding to an αi is denoted by si. The elements s1,…,sn form a set of generators for W. For w∈W denote the length of a minimal expression of w via the generators by l(w). We have a partial ordering on W called the Bruhat ordering. w′≤w, if there exists a reduced expression for w′ that can be obtained from a reduced expression for w by deleting a number of simple reflections.The monoidBr+ with generators {Tw,w∈W} and relations
(1)Tw1Tw2=Tw1w2ifl(w1)+l(w2)=l(w1w2)inW
is called the braid monoid of G.


### 2.2. Categories of Representations

For an algebraic groupH, we denote the Hopf algebra of polynomial functions on H by O(H). Let Rep(H) be the category of O(H)-comodules. This is an Abelian tensor category.LetPi be the parabolic subgroup of G containing B whose Levi subgroup has the root system {αi,-αi}. Using the natural Hopf algebra maps O(G)→O(B) and O(Pi)→O(B) we can get restriction functors
(2)Resi:Rep(Pi)⟶Rep(B),Res:Rep(G)⟶Rep(B).
The restriction functors are exact and naturally commute with taking tensor product of representations. Let H be a subgroup of G and M∈Rep(G). Define the H-invariant part of M to be MH:=HomRep(H)(k,M). Consider the induction functors
(3)Indi:Rep(B)⟶Rep(Pi),M⟼(O(Pi)⊗M)B,Ind:Rep(B)⟶Rep(G),M⟼(O(G)⊗M)B.
Set Δi:=Resi∘Indi↻Rep(B) and Δ:=Res∘Ind↻Rep(B). Notice that Δi and Δ are left exact, since the induction functors are left exact.


### 2.3. The Derived Categories

For an algebraic groupH, the regular comodule O(H) is injective in Rep(H); moreover, for any M∈Rep(H) the coaction map M→O(H)⊗M provides an embedding of M into an injective object. In particular, Rep(H) has enough injectives. The algebraic De Rham complex Ω•(H) provides an injective resolution for the trivial comodule, of the length equal to the dimension of H. For any M∈Rep(H) the complex Ω•(H)⊗M provides an injective resolution for M of the same length.Consider now the bounded derived categoriesDb(Rep(B)), Db(Rep(Pi)), and Db(Rep(G)). Let Li and L be the derived functors of Resi and Res, respectively. Denote the right derived functors of Indi and Ind by Ii and I, respectively. Let Di=Li∘Ii and D=L∘I be the right derived functors of Δi and Δ, respectively.Proposition 1.
(a) The functors Li and L are left adjoint to Ii and I, respectively.
(b)   For M∈Db(Rep(B)) and N∈Db(Rep(Pi)) (resp., for M∈Db(Rep(B)) and N∈Db(Rep(G))) we have the tensor identities:
(4)Ii(M⊗Li(N))≃Ii(M)⊗N,(resp.,I(M⊗L(N))≃I(M)⊗N).
(c) The functors Ii and I take the trivial O(B)-comodule to the trivial O(Pi)-comodule (resp., to the trivial O(G)-comodule).
(d)   Di and D are comonads for which the comonad maps Di→Di2 and D→D2 are isomorphisms.Proof.
The statements corresponding to (a) and (b) for Res and Ind (resp.,Resi and Indi) are Propositions 3.4 and 3.6 in [4]. The derived functors of a pair of adjoint functors are adjoint. (b) also follows from these statement for the non-derived functors since tensoring over a field is exact. 									(5)Ii(Id⊗Li)≃R(Indi(Id⊗Resi))≃R(Indi⊗Id)≃Ii⊗Id.
By (a) Di=Li∘Ii and D=L∘I are comonads (see [5, Section VI.1]). (b) and (c) imply that Ii∘Li(N)≃N for N∈Db(Rep(Pi)) and I∘L(N)≃N for N∈Db(Rep(G)). Thus, Id→~Ii∘Li (resp., Id→~I∘L) and from this we get the desired isomorphism
(6)Di=Li∘Ii=Li∘Id∘Ii⟶~Li∘Ii∘Li∘Ii=Di2.
and likewise for D.Remark 2.
It follows that the restriction functorsLi and L are fully faithful.


## 2.1. Root Data

LetG be a reductive algebraic group over an algebraically closed field k of characteristic zero. Let T be a Cartan subgroup of G and let (I,X,Y) be the corresponding root data, where I is the set of vertices of the Dynkin diagram, X is the weight lattice of G, and Y is the coroot lattice of G. Choose a Borel subgroup T⊂B⊂G. Denote the set of roots for G by Φ=Φ+⊔Φ-. Let {α1,…,αn} be the set of simple roots. The Weyl group W=Norm(T)/T of the fixed maximal torus acts naturally on the lattices X and Y and on the R-vector spaces spanned by them, by reflections in root hyperplanes. The simple reflection corresponding to an αi is denoted by si. The elements s1,…,sn form a set of generators for W. For w∈W denote the length of a minimal expression of w via the generators by l(w). We have a partial ordering on W called the Bruhat ordering. w′≤w, if there exists a reduced expression for w′ that can be obtained from a reduced expression for w by deleting a number of simple reflections.The monoidBr+ with generators {Tw,w∈W} and relations
(1)Tw1Tw2=Tw1w2ifl(w1)+l(w2)=l(w1w2)inW
is called the braid monoid of G.


## 2.2. Categories of Representations

For an algebraic groupH, we denote the Hopf algebra of polynomial functions on H by O(H). Let Rep(H) be the category of O(H)-comodules. This is an Abelian tensor category.LetPi be the parabolic subgroup of G containing B whose Levi subgroup has the root system {αi,-αi}. Using the natural Hopf algebra maps O(G)→O(B) and O(Pi)→O(B) we can get restriction functors
(2)Resi:Rep(Pi)⟶Rep(B),Res:Rep(G)⟶Rep(B).
The restriction functors are exact and naturally commute with taking tensor product of representations. Let H be a subgroup of G and M∈Rep(G). Define the H-invariant part of M to be MH:=HomRep(H)(k,M). Consider the induction functors
(3)Indi:Rep(B)⟶Rep(Pi),M⟼(O(Pi)⊗M)B,Ind:Rep(B)⟶Rep(G),M⟼(O(G)⊗M)B.
Set Δi:=Resi∘Indi↻Rep(B) and Δ:=Res∘Ind↻Rep(B). Notice that Δi and Δ are left exact, since the induction functors are left exact.


## 2.3. The Derived Categories

For an algebraic groupH, the regular comodule O(H) is injective in Rep(H); moreover, for any M∈Rep(H) the coaction map M→O(H)⊗M provides an embedding of M into an injective object. In particular, Rep(H) has enough injectives. The algebraic De Rham complex Ω•(H) provides an injective resolution for the trivial comodule, of the length equal to the dimension of H. For any M∈Rep(H) the complex Ω•(H)⊗M provides an injective resolution for M of the same length.Consider now the bounded derived categoriesDb(Rep(B)), Db(Rep(Pi)), and Db(Rep(G)). Let Li and L be the derived functors of Resi and Res, respectively. Denote the right derived functors of Indi and Ind by Ii and I, respectively. Let Di=Li∘Ii and D=L∘I be the right derived functors of Δi and Δ, respectively.Proposition 1.
(a) The functors Li and L are left adjoint to Ii and I, respectively.
(b)   For M∈Db(Rep(B)) and N∈Db(Rep(Pi)) (resp., for M∈Db(Rep(B)) and N∈Db(Rep(G))) we have the tensor identities:
(4)Ii(M⊗Li(N))≃Ii(M)⊗N,(resp.,I(M⊗L(N))≃I(M)⊗N).
(c) The functors Ii and I take the trivial O(B)-comodule to the trivial O(Pi)-comodule (resp., to the trivial O(G)-comodule).
(d)   Di and D are comonads for which the comonad maps Di→Di2 and D→D2 are isomorphisms.Proof.
The statements corresponding to (a) and (b) for Res and Ind (resp.,Resi and Indi) are Propositions 3.4 and 3.6 in [4]. The derived functors of a pair of adjoint functors are adjoint. (b) also follows from these statement for the non-derived functors since tensoring over a field is exact. 									(5)Ii(Id⊗Li)≃R(Indi(Id⊗Resi))≃R(Indi⊗Id)≃Ii⊗Id.
By (a) Di=Li∘Ii and D=L∘I are comonads (see [5, Section VI.1]). (b) and (c) imply that Ii∘Li(N)≃N for N∈Db(Rep(Pi)) and I∘L(N)≃N for N∈Db(Rep(G)). Thus, Id→~Ii∘Li (resp., Id→~I∘L) and from this we get the desired isomorphism
(6)Di=Li∘Ii=Li∘Id∘Ii⟶~Li∘Ii∘Li∘Ii=Di2.
and likewise for D.Remark 2.
It follows that the restriction functorsLi and L are fully faithful.


## 3. Demazure Descent

Fix a root data(I,X,Y) of the finite type, with the Weyl group W and the braid monoid Br+. Consider a triangulated category C.Definition 3.
A weak braid monoid action on the categoryC is a collection of triangulated functors
(7)Dw:C⟶C,w∈W
satisfying braid monoid relations; that is, for all w1,w2∈W there exist isomorphisms of functors
(8)Dw1∘Dw2≃Dw1w2,ifl(w1w2)=l(w1)+l(w2).Notice that we neither fix the braid relations isomorphisms nor impose any additional relations on them.Definition 4.
Demazure descent data on the categoryC is a weak braid monoid action {Dw} such that for each simple root si the corresponding functor Dsi is a comonad for which the comonad map Dsi→Dsi2 is an isomorphism.Here is the central construction of the paper. Consider a triangulated categoryC with a fixed Demazure descent data {Dw,w∈W} of the type (I,X,Y).Definition 5.
The descent categoryDesc(C,Dw,w∈W) is the full subcategory in C consisting of objects M such that for all i the cones of the counit maps Dsi(M)→ϵM are isomorphic to 0.Remark 6.
Suppose thatC has functorial cones. Then Desc(C,Dw,w∈W) a full triangulated subcategory in C being the intersection of kernels of Cone(Dsi→Id). However, one can prove this statement not using functoriality of cones.Lemma 7.
An objectM∈Desc(C,Dw,w∈W) is naturally a comodule over each Dsi.Proof.
By definition the comonad maps(9)η:Dsi⟶Dsi2,ϵ:Dsi⟶Id
make the following diagram commutative:
(10)
For Demazure descent data we require that η is an isomorphism, so ϵ∘Dsi is also an isomorphism. Let M∈Desc(C,Dw,w∈W). That Cone(Dsi(M)→ϵM) is isomorphic to 0 is equivalent to saying that Dsi(M)→ϵM is an isomorphism. This gives the commutative diagram.
Consider(11)
Thus, ϵ-1 satisfies the axiom for the coaction.Remark 8.
Recall that in the usual descent setting either in Algebraic Geometry or in abstract Category Theory (Barr-Beck theorem) descent data includes a pair of adjoint functors and their composition which is a comonad. By definition, the descent category for such data is the category of comodules over this comonad. Our definition ofDesc(C,Dw,w∈W) for Demazure descent data formally is not about comodules, yet the previous Lemma demonstrates that every object of Desc(C,Dw,w∈W) is naturally equipped with structures of a comodule over each Di and any morphism in Desc(C,Dw,w∈W) is a morphism of Di-comodules.


## 4. Main Theorem

We now go back to consideringDi=Li∘Ii and D=L∘I.Proposition 9.
Letw∈W and let w=si1⋯sin be a reduced expression. Then Dw:=Di1∘⋯∘Din is independent of the choice of reduced expression and the Dw's form Demazure descent data on C=Db(Rep(B)).Lemma 10.
Letw=si1⋯sin be a reduced expression. Then
(12)Pi1⋯Pin=⋃w′≤w‍Bw′B,
where the union is over all w′∈W which is ≤w in the Bruhat order.Proof.
The proof goes by induction onn=l(w). It is true for n=1 by definition of Pi. Set v=si1⋯sin-1. Using the hypotheses we get
(13)Pi1⋯Pin-1Pin=(⋃w′≤v‍Bw′B)(B∪BsinB)=⋃w′≤v‍Bw′B∪⋃w′≤v‍(Bw′B)(BsinB).
Let w′ be any element in W and s a simple reflection. Then by [6, Corollary 28.3] we have (Bw′B)(BsB)⊆Bw′sB∪Bw′B. Thus, if w′sin≤w′≤v, then (Bw′B)(BsinB) is contained in the first union. If w′≤w′sin, then we have (Bw′B)(BsinB)=Bw′sinB by [6, Lemma 29.3A and section 29.1]. Thus, the product can be written as
(14)Pi1⋯Pin=⋃w′≤v‍Bw′B∪⋃w′≤vw′≤w′sin‍Bw′sinB=⋃w′≤v‍Bw′B∪⋃w′′sin≤v,w′′sin≤w′′‍Bw′′B.
Claim. The conditions w′′sin≤v and w′′sin≤w′′ are equivalent to the conditions w′′≤w and w′′sin≤w′′.
Proof of the Claim. Assume that w′′sin≤v. By [7, Proposition 5.9] this implies that w′′≤v or w′′≤vsin=w. In both cases we get w′′≤w since v≤w. Assume now that w′′≤w and w′′sin≤w′′. w′′ has a reduced expression of the form
(15)w′′=si1⋯s^ij1⋯s^ij2⋯s^ijk⋯sin,
where the   ^   indicates that the term has been removed from the product. If jk≠n, then
(16)w′′sin=si1⋯s^ij1⋯s^ij2⋯s^ijk⋯sin-1≤si1⋯sin-1=v.
If jk=n, then w′′≤v. Since w′′sin≤w′′ by assumption we get w′′sin≤v.
This completes the proof of the claim.
Ifw′≤v in the first union satisfies that w′sin≤w′, then it is also contained in the second union. Using the claim we get
(17)Pi1⋯Pin=⋃w′≤vw′≤w′sin‍Bw′B∪⋃w′′≤w,w′′sin≤w′′‍Bw′′B.
Assume that w′≤w and w′≤w′sin. Then w′ has a reduced expression of the form
(18)w′=si1⋯s^ij1⋯s^ij2⋯s^ijk⋯sin.
If jk=n, then w′≤v. If jk≠n, then w′sin≤v, but since w′≤w′sin we get w′≤v. Hence, the conditions w′≤v and w′≤w′sin can be replaced by w′≤w and w′≤w′sin. Thus,
(19)Pi1⋯Pin=⋃w′≤ww′≤w′sin‍Bw′B∪⋃w′′≤w,w′′sin≤w′′‍Bw′′B=⋃w′≤w‍Bw′B.
This finishes the induction step.Proof of the Proposition.
Letw∈W and let si1⋯sin=sji⋯sjn be two reduced expressions for w. By Lemma 10 this implies that Pi1⋯Pin=Pj1⋯Pjn. By [8, Theorem 3.1] the B-module structure of Δi1∘⋯∘Δin is determined upto a natural isomorphism by the set Pi1⋯Pin. Hence
(20)Δi1∘⋯∘Δin≃Δj1∘⋯∘Δjn.
Hence, for any choice of reduced expression we can define
(21)Δw:=Δi1∘⋯∘Δim.
Let w1 and w2 be elements in W such that l(w1w2)=l(w1)+l(w2). Pick reduced expressions si1⋯sir and sj1⋯sjt for w1 and w2, respectively. Then si1⋯sirsj1⋯sjt is a reduced expression for w1w2 and we get braid relations for the Δw(22)Δw1∘Δw2=Δi1∘⋯∘Δir∘Δj1∘⋯∘Δjt=Δw1w2.
Define
(23)Dw≔R(Δw)=R(Δi1∘⋯∘Δim)=R(Δi1)∘⋯∘R(Δim)=Di1∘⋯∘Dim.
The braid relations for Dw now follows from the braid relations for Δw:
(24)Dw1∘Dw2=R(Δw1)∘R(Δw2)≃R(Δw1∘Δw2)≃R(Δw1w2)=Dw1w2.Theorem 11.
D
e
s
c
(
C
,
D
w
,
w
∈
W
) is equivalent to Db(Rep(G)).Proof.
LetM∈Db(Rep(B)). Being able to extend M to an element in Db(Rep(G) is equivalent to M being in the image of L. Assume M=L(N) for some N∈Db(Rep(G)). Then D(M)=L∘I∘L(N)≃L(N)=M. If D(M)→~M, then M≃L(I(M)), so M is in the image of L. Thus, being in the image of L is equivalent to D(M)→M being an isomorphism which is again equivalent to M∈ker⁡(C), where C:=Cone(D→Id). Set Ci:=Cone(Di→Id).
Claim. ker⁡(C)=⋂i‍ker⁡(Ci).
Proof of Claim. Assume that M∈ker⁡(C). Then M=L(N) for some N∈Db(Rep(G)). But then M=Li(N|Pi) for all i, so Di(M)→M is an isomorphism for all i. Hence, M∈⋂i‍ker⁡(Ci). Assume that M∈⋂iker⁡(Ci). Then all Di(M)→M are isomorphisms. Choose a reduced expression si1⋯siN for the longest element in the Weyl group. Then Pi1⋯PiN=G. By [8] we have D=Di1∘⋯∘DiN. 							(25)D(M)≃Di1∘⋯∘DiN(M)≃Di1∘⋯∘DiN-1(M)≃⋯≃Di1(M)≃M.
Hence,
(26)Cone(D(M)⟶M)≃Cone(D(D(M))⟶D(M)).
By definition of a comonad we have the following commutative diagram:
(27)
Since η is an isomorphism so is ϵD and thus Cone(D(D(M))→D(M))=0. This shows that M∈ker⁡(C).
This completes the proof of the claim.
From the claim we get that(28)Db(Rep(G))=⋂i‍ker⁡(Ci),
which is exactly the descent category.


## 5. Further Directions

### 5.1. Quantum Groups

Fix a root data(I,X,Y) of the finite type. Let  UA be the Lusztig quantum group over the ring of quantum integers A=Z[v,v-1]. Denote the quantum Borel subalgebra by  BA. For a simple root αi the corresponding quantum parabolic subalgebra is denoted by  Pi,A.Following [9] we consider the categories of locally finite weight modules over  UA (resp., over  BA, resp., over  Pi,A) denoted by Rep(UA) (resp., by Rep(BA), resp., by Rep(Pi,A)). We consider the corresponding derived categories Db(Rep(UA)), Db(Rep(BA)), and Db(Rep(Pi,A)).Like in the reductive algebraic group case, the restriction functors(29)L:Db(Rep(UA))⟶Db(Rep(BA)),Li:Db(Rep(Pi,A))⟶Db(Rep(BA))
are fully faithful and possess right adjoint functors denoted by I (resp., by Ii). Denote the comonad Li∘Ii by Di. Andersen, Polo, and Wen proved that the functors Di define a weak braid monoid action on the category Db(Rep(BA)). One can easily prove that the functors form Demazure descent data. The corresponding descent category Desc(Db(Rep(BA)),D1,…Dn) is equivalent to Db(Rep(UA)).


### 5.2. Equivariant Sheaves

LetX be an affine scheme equipped with an action of a reductive algebraic group G. Fix a Borel subgroup B⊂G. Like in the main body of the present paper, consider the minimal parabolic subgroups in G denoted by P1,…,Pn. Denote the derived categories of quasicoherent sheaves on X equivariant with respect to G (resp., B, resp., Pi) by Db(QCohG(X)) (resp., by Db(QCohB(X)), resp., by Db(QCohPi(X))). We have the natural functors provided by restriction of equivariance L:Db(QCohG(X))→Db(QCohB(X)) and Li:Db(QCohPi(X))→Db(QCohB(X)). These functors have the right adjoint ones I, resp. I1,…In. The comonads D1,…Dn given by the compositions of extension and restriction of equivariance define a Demazure descent data on the category Db(QCohB(X)). The corresponding descent category is equivalent to Db(QCohG(X)).


### 5.3. Algebraic Loop Group

For a simple algebraic groupG consider the algebraic loop group LG=Map(D•,G) (resp., the formal arcs group L+G=Map(D,G)). Here D (resp., D•) denotes the formal disc (resp., the formal punctured disc). Consider the affine Kac-Moody central extension
(30)1⟶Gm⟶LG^⟶LG⟶1.
The affine analog of the Borel subgroup B⊂G is the Iwahori subgroup Iw⊂L+G. Let P0,…,Pn be the standard minimal parahoric subgroups in L+G. One considers the adjoint pairs of coinduction-restriction functors I0,L0,…,In,Ln between Db(Rep(Iw)) and Db(Rep(Pi)). Denote the comonads Li∘Ii by Di for i=0,…,n. We claim that D0,…,Dn form affine Demazure descent data on Db(Rep(Iw)). We conjecture that the descent category is equivalent to Db(Rep(LG^)) (direct sum of the categories over all positive integral levels).


## 5.1. Quantum Groups

Fix a root data(I,X,Y) of the finite type. Let  UA be the Lusztig quantum group over the ring of quantum integers A=Z[v,v-1]. Denote the quantum Borel subalgebra by  BA. For a simple root αi the corresponding quantum parabolic subalgebra is denoted by  Pi,A.Following [9] we consider the categories of locally finite weight modules over  UA (resp., over  BA, resp., over  Pi,A) denoted by Rep(UA) (resp., by Rep(BA), resp., by Rep(Pi,A)). We consider the corresponding derived categories Db(Rep(UA)), Db(Rep(BA)), and Db(Rep(Pi,A)).Like in the reductive algebraic group case, the restriction functors(29)L:Db(Rep(UA))⟶Db(Rep(BA)),Li:Db(Rep(Pi,A))⟶Db(Rep(BA))
are fully faithful and possess right adjoint functors denoted by I (resp., by Ii). Denote the comonad Li∘Ii by Di. Andersen, Polo, and Wen proved that the functors Di define a weak braid monoid action on the category Db(Rep(BA)). One can easily prove that the functors form Demazure descent data. The corresponding descent category Desc(Db(Rep(BA)),D1,…Dn) is equivalent to Db(Rep(UA)).


## 5.2. Equivariant Sheaves

LetX be an affine scheme equipped with an action of a reductive algebraic group G. Fix a Borel subgroup B⊂G. Like in the main body of the present paper, consider the minimal parabolic subgroups in G denoted by P1,…,Pn. Denote the derived categories of quasicoherent sheaves on X equivariant with respect to G (resp., B, resp., Pi) by Db(QCohG(X)) (resp., by Db(QCohB(X)), resp., by Db(QCohPi(X))). We have the natural functors provided by restriction of equivariance L:Db(QCohG(X))→Db(QCohB(X)) and Li:Db(QCohPi(X))→Db(QCohB(X)). These functors have the right adjoint ones I, resp. I1,…In. The comonads D1,…Dn given by the compositions of extension and restriction of equivariance define a Demazure descent data on the category Db(QCohB(X)). The corresponding descent category is equivalent to Db(QCohG(X)).


## 5.3. Algebraic Loop Group

For a simple algebraic groupG consider the algebraic loop group LG=Map(D•,G) (resp., the formal arcs group L+G=Map(D,G)). Here D (resp., D•) denotes the formal disc (resp., the formal punctured disc). Consider the affine Kac-Moody central extension
(30)1⟶Gm⟶LG^⟶LG⟶1.
The affine analog of the Borel subgroup B⊂G is the Iwahori subgroup Iw⊂L+G. Let P0,…,Pn be the standard minimal parahoric subgroups in L+G. One considers the adjoint pairs of coinduction-restriction functors I0,L0,…,In,Ln between Db(Rep(Iw)) and Db(Rep(Pi)). Denote the comonads Li∘Ii by Di for i=0,…,n. We claim that D0,…,Dn form affine Demazure descent data on Db(Rep(Iw)). We conjecture that the descent category is equivalent to Db(Rep(LG^)) (direct sum of the categories over all positive integral levels).



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*Source: 823467-2014-05-25.xml*