Affine Springer fibers and the representation theory
of small quantum groups and related algebras
by
Pablo Boixeda Alvarez
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2020
c
○ Massachusetts Institute of Technology 2020. All rights reserved.
Author . . . .
Department of Mathematics
April 23, 2020
Certified by . . . .
Roman Bezrukavnikov
Professor of Mathematics
Thesis Supervisor
Accepted by . . . .
Wei Zhang
Chairman, Department Committee on Graduate Theses
Affine Springer fibers and the representation theory of small
quantum groups and related algebras
by
Pablo Boixeda Alvarez
Submitted to the Department of Mathematics on April 23, 2020, in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
Abstract
This thesis deals with the connections of Geometry and Representation Theory. In particular we study the representation theory of small quantum groups and Frobenius kernels and the geometry of an equivalued affine Springer fiber ℱ 𝑙𝑡𝑠 for 𝑠 a regular
semisimple element. In Chapter 2 we relate the center of the small quantum group with the cohomology of the above affine Springer fiber. This includes joint work with Bezrukavnikov, Shan and Vaserot. In Chapter 3 we study the geometry of the affine Springer fiber and in particular understand the fixed points of a torus action contained in each component. In Chapter 4 we further have a collection of algebraic results on the representation theory of Frobenius kernels. In particular we state some results pointing towards some construction of certain partial Verma functors and we compute this in the case of 𝑆𝐿2. We also compute the center of Frobenius kernels
in the case of 𝑆𝐿2 and state a conjecture on a possible inductive construction of the
general center.
Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics
Acknowledgments
I want to thank my advisor, Roman Bezrukavnikov, for all that he has taught me and the support throughout the last five years. I have also greatly benefited from discussions with many other mathematicians. I want to thank Pavel Etingof, Zhiwei Yun, Ivan Losev, Geordie Williamson, Viet Bao Le Hung, Michael McBreen, Chris Ryba, Guangyi Yue, Dimitrii Kubrak. I have learned a lot of mathematics from all of you. I also want to thank the mathematicians that taught and inspired me, before coming to MIT: Ian Grojnowski, Andras Zsak and Albrecht Hess. Finally I want to thank my parents Juan Pablo Boixeda and Marta Alvarez for always supporting me, my grandparents Placido Alvarez, Clotilde Holgado and Ricardo Boixeda for always believing in me and Ana Novak, I could not have done this without your love and support.
Contents
1 Introdution 9
2 On the center of the small quantum group 13
2.1 Introduction . . . 13
2.1.1 Notation . . . 14
2.1.2 Overview of results . . . 21
2.1.3 Outline of the Chapter . . . 21
2.2 Equivariant cohomology and GKM . . . 22
2.3 Basics of the representation theory of the small quantum group . . . 24
2.4 The representation theory for sl2 . . . 26
2.5 The map from 𝐻*(ℱ 𝑙𝑒) . . . 28
2.6 Map from 𝐻*(ℱ 𝑙) . . . 33
2.7 A deformation theory setting . . . 36
2.8 The center of the flat deformation . . . 37
2.9 The compatibility of both maps to the center . . . 43
2.10 The vanishing of certain generalized 𝐻𝐻1 . . . . 47
2.11 A deformation theory argument in the equivariant setting . . . 51
2.12 Algebraic interpretation of the map from 𝐻*(ℱ 𝑙) . . . 55
2.13 Compatibility of the maps from 𝐻*(ℱ 𝑙) and 𝐻*(ℱ 𝑙𝑒) . . . 61
3 Fixed Points and Components of equivalued affine Springer fibers 71 3.1 Introduction . . . 71
3.2 Components of equivalued affine Springer fibers . . . 73
3.3 Fixed points of components . . . 77
3.3.1 Neighborhoods of fixed points . . . 77
3.3.2 Fixed points of components in Type A . . . 78
4 Toward partial Verma functors of 𝒰[𝑟](g) and related results 81 4.1 Introduction . . . 81
4.1.1 Organization of the Chapter . . . 82
4.2 Steinberg Tensor Product Theorem for 𝒰[𝑟](g) . . . 82
4.3 Towards partial Verma functors . . . 85
4.4 Computations for 𝑆𝑙2 . . . 89
4.4.1 Presentation of the algebra . . . 89
4.4.2 Equivalence in the case of 𝑆𝑙2 . . . 96
4.4.3 Center of Frobenius kernels of 𝑆𝐿2 . . . 99
4.4.4 Conjectures on the construction of the center in general . . . . 102
A Combinatorial lemmas of Chapter 3 103 B Fixed points and closures 107 B.1 Neighborhoods of fixed points in type A . . . 107
B.2 Computing open parts of components in type A . . . 108
B.3 Determinants and fixed points . . . 110
B.4 Degree maximization algorithm . . . 114
Chapter 1
Introdution
This thesis contains 3 parts.
There are two main settings, one algebraic and one geometric.
The first is the algebraic setting. This is the representation theory of the small tum group and of the Frobenius kernel. The representation theory of the small quan-tum group and the first Frobenius kernel are related by work of Andersen, Jantzen and Soergel [1].
The second setting is a geometric setting. This will be given by certain affine Springer fibers known as equivalued Springer fibers. These are the affine Springer ℱ 𝑙𝑒𝑘 fibers
corresponding to the element 𝑒𝑘 := 𝑡𝑘𝑠 ∈ g((𝑡)), where 𝑠 ∈ g is a regular
semisim-ple element. In these projects we focus on the affine Springer fiber corresponding to 𝑒 := 𝑒1 = 𝑡𝑠. These affine Springer fibers have been studied before by several authors.
The main results we use are from work of Goresky, Kottwitz and MacPherson [26] [27] [28]. Other authors who have studied this affine Springer fiber are Lusztig [42], Hikita [30] and Kivinen [36].
The three parts in this thesis study either one of the settings or a relation between the two.
In Chapter 2 we study the question of the center of the small quantum group. This will be part of an upcoming paper with Bezrukavnikov, Shan and Vasserot [10]. In particular we study the center of a regular block. This question has been studied before in particular by work of Bezrukavnikov, Lachowska and Qi [14] [37] [38] [39].
They give a description of the center in terms of the cohomology of certain coherent sheaves on the Springer resolution. In this project we will give an approach at the description of the center in terms of the cohomology of the above affine Springer fiber ℱ 𝑙𝑒 for the Langlands dual group 𝐺∨. There is an action by the adjoint group of
the corresponding Lie algebra on this center and we will study the 𝐺-invariant and 𝑇 -invariant parts of this center, where 𝑇 is a maximal torus, whose Lie Algebra con-tains 𝑠.
This project thus joins both settings described above. In upcoming work of Bezrukavnikov and McBreen [15], they give a connection between a certain category of graded mod-ules of the small quantum group in terms of certain microlocal sheaves on the affine Springer fiber, further establishing the connection between the above algebraic and geometric settings.
The main result in the project is the following commutative diagram
𝐻*(ℱ 𝑙) 𝑍(𝑢^0 𝑞− 𝑚𝑜𝑑)𝐺 𝐻*(ℱ 𝑙𝑒)Λ 𝑍(𝑢 ^ 0 𝑞 − 𝑚𝑜𝑑)𝑇 .
We further prove the injectivity of the bottom map. The surjectivity of the top map remains a conjecture, but in the project we introduce a deformation theory setting, that could be used to prove surjectivity and we prove this center lifts to the first order deformation. The lifting to higher order deformations still remains an open question. In Chapter 3 we study the geometry of the affine Springer fiber ℱ 𝑙𝑒 in Type A. The
affine Springer fiber has an action of a maximal torus 𝑇 . In this project we study the fixed points contained in each component. In particular we determine a subset of the components classified by the fundamental box 𝐹 , a subset of the affine Weyl group corresponding to the subset of alcoves given by {0 ≤< 𝜆, 𝛼𝑖 >≤ 1} for 𝛼𝑖 the finite
simple roots, under the bijection of the affine Weyl group with the set of alcoves. Here the bijection is given by the identity corresponding to the fundamental alcove {0 ≤< 𝜆, 𝛼 >≤ 1} for all positive roots 𝛼.
We denote 𝑌𝑤 for 𝑤 ∈ 𝐹 for the corresponding component. We will understand the
fixed points of 𝑌𝑤. This is enough to understand the fixed points of every component
using translations and central symmetries of the affine Springer fiber. The result is then the following
𝑌𝑤𝑇 = {𝑤′ ≤ 𝑤0𝑤}
where the order is the Bruhat order on the affine Weyl group.
The equivalent result is still unknown for the finite Springer fibers so this is a par-ticular result for the above setting. It has application to Representation Theory by work of Le Hung. In joint work of the author with Guangyi Yue, this can also be related to the combinatorics of affine RSK as constructed by Chmutov, Pylyavskyy, Yudovina [19] and the combinatorics of two sided cells as described by Lusztig [43] [44]. This thus answers a conjecture by Lusztig [42] in this particular case in Type A.
The analogous results in other types should conjecturally also be true, but remains an open question.
In Chapter 4 we will study the representation theory of certain algebras 𝒰[𝑟](g), which is the characteristic 𝑝 reduction of certain integral forms of the enveloping algebra. These can be seen as certain flat central deformations of the 𝑟th Frobenius kernel in a similar way as the enveloping algebra 𝒰 (g) can be understood as such for the first Frobenius kernel.
In this chapter we have a collection of results relating to these algebras. We start by generalizing the Steinberg tensor product theorem to this algebra.
In the second section we relate the representation theory of 𝒰[𝑟](g) at the 0 𝑝-center
parameter, ie the representation theory of the 𝑟th Frobenius kernel, with the represen-tation theory of 𝒰[𝑟+1](g) at a generic semisimple character. This is a generalization
of a relation between the representation theory of 𝒰 (g) at a 𝑝-character 𝜒, whose stabilizer Levi is 𝐿 and the representation theory of 𝒰 (l) corresponding to this Levi. This relation points towards certain partial Verma functor construction.
We end the chapter with a further study of 𝑆𝐿2. We first prove that the relation
discussed in the previous section is an equivalence in this case and we end with a computation of the center of Frobenius kernels in this case. We then state a conjec-tural construction of the center of the Frobenius kernels for other types.
Chapter 2
On the center of the small quantum
group
2.1
Introduction
The goal of this chapter is to give a geometric description of the center of the small quantum group in terms of the cohomology of a certain affine Springer fiber of the Langlands dual group.
This adds on to work by Bezrukavnikov, Lachowska and Qi [14] [37] [38] [39], that have different approaches to understanding the center of this algebra, by relating it to the cohomology of certain coherent sheaves on the Springer resolution.
In this approach we will use several interpretations of Representation theoretical cat-egories in terms of coherent sheaves on certain algebraic varieties and some catcat-egories of perverse sheaves in certain affine flag varieties and partial flag varieties. This will rely on work by Arkhipov, Bezrukavnikov, Braverman, Finkelberg, Gaitsgory, Ginzburg, Mircović, Yun [2] [4] [16] [24] [3] [9] [7] [11] [12] and many other authors. We will relate it to the cohomology of a certain affine Springer fiber, that has been studied before by several authors.
There are some basic results following work by Goresky, Kottwitz and MacPherson [28], relating to this Springer fiber and basic properties of its cohomology. We will use this result in this chapter to give a description of the center of a certain algebra,
which will induce a map from the required cohomology to the center of a regular block of the small quantum group.
There is further work about this affine Springer fiber. Expanding the combinatorial description given by GKM, we have work by Hikita and by Kivinen. In upcoming joint work with Losev and an Appendix with Kivinen, the combinatorial descriptions are further expanded and this cohomology in Type A is related to the geometry of Hilbert schemes on C2.
Further work understanding the geometry is given by work of Lusztig [42]. In the following chapter and upcoming joint work with Yue, the geometry of the compo-nents of this space are further understood as well as relating the geometry of these components to the combinatorics of 2-sided cells introduced by Lusztig.
In upcoming work of Bezrukavnikov and McBreen [15], the representation theory of the small quantum group and the geometry of the above affine Springer fiber are further related, by given a Koszul duality relation between a category of graded mod-ules of the small quanutm group and a category of microlocal sheaves on the affine Springer fiber.
2.1.1
Notation
We start by introducing several algebraic and geometric objects that we will use throughout the chapter.
Let 𝐺 be a semisimple algebraic group. Denote by g its Lie algebra. Similarly we will denote by 𝐵 a Borel and b its Lie algebra. We will let 𝑁 be the unipotent radical of 𝐵 and n the nilpotent radical of b.
We introduce 𝑊 the finite Weyl group of 𝐺. Let (Λ, 𝑅, Λ*, 𝑅*) the root system of the above group. Let also ̃︁𝑊 , be the affine Weyl group. Further we denote by 𝑓𝑊 ,̃︁ the set of minimal elements in the 𝑊 cosets in 𝑊 ∖̃︁𝑊 .
We will denote by 𝐺𝑣𝑒𝑒 the Langlands dual group, ie the groups with root system given by (Λ*, 𝑅*, Λ, 𝑅) and similar notation for their Borel and unipotent radical of the Borel.
With this notation we introduce some algebraic varieties that will appear in the proof. We let ˜g := {(b, 𝑥)|𝑥 ∈ b ∈ 𝐺/𝐵} be the Grothendieck-Springer resolution and similarly ˜𝒩 := {(b, 𝑥)|𝑥 ∈ n, b ∈ 𝐺/𝐵} be the Springer resolution. These have obvious actions of the group 𝐺 and equivariant maps ˜g→ g and ˜𝒩 → g, where the last map has image in nilpotent elements.
We can further introduce the affine flag variety and several categories related to it. Denote by 𝒦 := F((𝑡)) and 𝒪 := F[[𝑡]], where F is the algebraic closure of F𝑝 for some
prime 𝑝. We will need to consider certain Artin-Schreier sheaves and that is why we consider the following geometric over a field of finite characteristic.
Now we can introduce the loop group 𝐺∨(𝒦). We consider an Iwahori subgroup 𝐼 of this, defined by the pullback diagram
𝐼 𝐺∨(𝒪)
𝐵∨ 𝐺∨
We use this to define the following ind-schemes. The affine flag variety is defined by ℱ 𝑙 := 𝐺∨(𝒦)/𝐼. This is well known to be an ind-projective ind-scheme and the left
𝐼-orbits on ℱ 𝑙 are finite dimensional and we can consider several categories of perverse sheaves. We can also consider 𝐼0 the unipotent radical of 𝐼 and we can consider the
𝑇 bundle over ℱ 𝑙, denoted by ̃︁ℱ 𝑙 := 𝐺∨(𝒦)/𝐼
0. Following the work of Bezrukavnikov
and Yun [16] we will be able to define several categories of perverse sheaves on ̃︁ℱ 𝑙 too. Finally we can also introduce the affine Grassmannian 𝒢𝑟 := 𝐺∨(𝒦)/𝐺∨(𝒪). We will now introduce some categories of perverse sheaves [6] of 𝑙′-adic sheaves, where 𝑙′ is a prime different from 𝑝 and some particular objects. Here we follow some con-structions done in Bezrukavnikov, Yun [16].
Consider the categories 𝑃𝐼𝐼 := 𝑃 𝑒𝑟𝑣𝐼(ℱ 𝑙), 𝑃𝐼0𝐼 := 𝑃 𝑒𝑟𝑣𝐼0(ℱ 𝑙), 𝑃𝐼0𝐼0 := 𝑃 𝑒𝑟𝑣(𝐼0∖̃︁ℱ 𝑙)
as defined in Bezrukavnikov, Yun [16], 𝑃𝐺(O)𝐼 := 𝑃 𝑒𝑟𝑣𝐺∨(𝒪)(ℱ 𝑙) and 𝑃𝐺(O)𝐼 0 :=
𝑃 𝑒𝑟𝑣𝐺∨(𝒪)(̃︁ℱ 𝑙) of ¯Q𝑙′ perverse sheaves. Also consider the Iwahori-Whittaker
cate-gories as considered by Bezrukavnikov, Braverman and Mircović [11]. To construct these consider the negative Iwahori 𝐼− corresponding to the opposite Borel. Further
consider a generic character 𝜓 : 𝐼0− → G𝑎, where 𝐼0− is the unipotent radical of 𝐼 −
. Then consider the Artin-Schreier sheaf 𝐴𝑆 on G𝑎. This is constructed as follows.
Consider the Artin-Schreier map 𝐴𝑆 := 𝐹 − 𝑖𝑑 : G𝑎 → G𝑎, where 𝐹 is the Frobenius.
This map is just given as the quotient of the additive group action of G𝑎(F𝑝). We
thus get the pushforward of the constant sheaf is just given by the sum of sheaves corresponding to each character of this group. Now let 𝐴𝑆 be the sheaf corresponding to a non-trivial character. Now we can consider the sheaf 𝜓*(𝐴𝑆) on 𝐼0−. Then we can consider sheaves on ℱ 𝑙 or on ̃︁ℱ 𝑙 that are equivariant with respect to the pair (𝐼0−, 𝜓). We will denote the perverse categories we get by 𝑃𝐼𝑊 𝐼 := 𝑃 𝑒𝑟𝑣𝐼𝑊(ℱ 𝑙) and
𝑃𝐼𝑊 𝐼0 := 𝑃 𝑒𝑟𝑣𝐼𝑊( ̃︀𝐹 𝑙).
We will also consider the triangulated categories corresponding to the above, which we will denote by 𝐷𝐼𝐼 := 𝐷𝐼(ℱ 𝑙), 𝐷𝐼𝐼0... Similarly we will denote the mixed version
by 𝐷𝑚,𝐼𝐼, 𝐷𝑚,𝐼𝐼0...
We now will introduce certain objects in these categories. For this we will classify orbits corresponding to 𝐼 and 𝐺∨(𝒪) on ℱ 𝑙.
Note that we have 𝑁 (𝑇 (𝒦))/𝑇 (𝒪) ∼= ̃︁𝑊 . So similarly to the finite Bruhat orbit, we get the following decomposition
𝐺∨(𝒦) = ⨿𝑤∈̃︁𝑊𝐼 ˙𝑤𝐼
for any choice of lifts ˙𝑤 of 𝑤. It thus follows that the 𝐼-orbits of ℱ 𝑙 are given by ̃︁𝑊 and we denote by 𝑗𝑤 : ℱ 𝑙𝑤 → ℱ 𝑙 the inclusion of the corresponding orbit. Note that
ℱ 𝑙𝑤 ∼= A𝑙(𝑤).
It now follows immediately that the 𝐺∨(𝒪)-orbits on ℱ 𝑙 are given by 𝑊 ∖̃︁𝑊 ∼= Λ. We will denote 𝑗𝑤! := (𝑗𝑤)!( ¯Q𝑙′[𝑙(𝑤)]) and 𝑗𝑤* := (𝑗𝑤)*( ¯Q𝑙′[𝑙(𝑤)]), the standard and
costandard objects in 𝑃𝐼𝐼 and denote by 𝐿𝑤 := 𝐼𝐶ℱ 𝑙¯𝑤 the simple object. In a slight
abuse of notation we will use the same notation for similarly defined objects in 𝑃𝐼0𝐼.
We will also use the notation ˜𝑗𝑤!, ˜𝑗𝑤*to be the free monodromic version of the above
in the category 𝑃𝐼0𝐼0. These objects are constructed in Bezrukavnikov, Yun [16].
and again by abuse of notation we will denote by the same the objects in 𝑃𝐺∨(𝒪)𝐼 0.
Now for the Iwahori-Whittaker categories, we note that the 𝐼0− orbits are also de-scribed by ̃︁𝑊 . But an orbit will have a non-zero object with support at that orbit if and only if the character 𝜓 is trivial on 𝑆𝑡𝑎𝑏𝐼−
0 (𝑤). This condition can be easily
checked to be equivalent to 𝑤 ∈𝑓̃︁𝑊 . Recall that we denote by 𝑓𝑊 , the elements of̃︁ ̃︁
𝑊 that are minimal in the corresponding left 𝑊 -orbit.
With this we can construct the objects ∆𝑤 := (𝑗𝑤)!(𝐴𝑆ℱ 𝑙−𝑤[𝑙(𝑤)]) and ∇𝑤 := (𝑗𝑤)*(𝐴𝑆ℱ 𝑙−𝑤[𝑙(𝑤)])
for 𝑤 ∈𝑓
̃︁
𝑊 . Here we denote by 𝐴𝑆ℱ 𝑙−
𝑤 the pullback of 𝐴𝑆 to the 𝐼
−
0 orbit ℱ 𝑙 − 𝑤 under
the map 𝜓 : 𝐼0− → G𝑎.
We further have functors between these categories, that we will use repeatedly. Consider the map of stacks
𝜋𝐺∨(𝒪) : 𝐼∖ℱ 𝑙 → 𝐺∨(𝒪)∖ℱ 𝑙
Similarly we will by abuse of notation use the notation
𝜋𝐺∨(𝒪) : 𝐼∖̃︁ℱ 𝑙 → 𝐺∨(𝒪)∖̃︁ℱ 𝑙
We can thus consider the functors
(𝜋𝐺∨(𝒪))* : 𝐷𝐼𝐼 → 𝐷𝐺∨(𝒪)𝐼 (𝜋𝐺∨(𝒪))* : 𝐷𝐼𝐼
0 → 𝐷𝐺∨(𝒪)𝐼0
(𝜋𝐺∨(𝒪))*, (𝜋𝐺∨(𝒪))!: 𝐷𝐺∨(𝒪)𝐼 → 𝐷𝐼𝐼 (𝜋𝐺∨(𝒪))*, (𝜋𝐺∨(𝒪))! : 𝐷𝐺∨(𝒪)𝐼
0 → 𝐷𝐼𝐼0
Similarly we can consider the averaging functors with respect to the character 𝜓 and we thus get the following set of functors
𝐴𝑣𝐼𝑊 : 𝐷𝐼0𝐼 → 𝐷𝐼𝑊 𝐼 𝐴𝑣𝐼𝑊 : 𝐷𝐼0𝐼0 → 𝐷𝐼𝑊 𝐼0
Lastly there are the functors
𝐷𝐼𝐼 → 𝐷𝐼0𝐼 𝐷𝐼0𝐼 → 𝐷𝐼𝐼
𝐷𝐼𝐼0 → 𝐷𝐼0𝐼0 𝐷𝐼0𝐼0 → 𝐷𝐼𝐼0
given by forgetful functors or averaging/pushforward functors.
These categories and functors are related by Koszul duality by work of Bezrukavnikov, Yun [16]. We will use these results throughout the chapter.
We now introduce the affine Springer fiber that will be relevant in this chapter. To define this note that for 𝑒 ∈ g∨((𝑡)) we can define the affine Springer fiber as
ℱ 𝑙𝑒 := {𝑔𝐼|𝑒 ∈ 𝑔𝐿𝑖𝑒(𝐼)𝑔−1}.
In this chapter we will consider the affine Springer fiber for the element 𝑒 := 𝑡𝑠 for 𝑠 ∈ g∨ a regular semisimple element in the constant Lie algebra g∨ ˓→ g∨((𝑡)). Fur-ther we can assume WLOG that 𝑠 lies in t∨ the Lie algebra of the above maximal torus.
We can consider the action of 𝐺∨(𝒦) by left multiplication on ℱ 𝑙 and restrict this to 𝑇∨. This action preserves ℱ 𝑙𝑒, as 𝑒 commutes with the action of 𝑇∨. The fixed points
of this action on ℱ 𝑙 are given by ̃︁𝑊 ˓→ ℱ 𝑙, the points considered above. Further all these points lie in ℱ 𝑙𝑒. We will use some work by Goresky, Kottwitz and MacPherson
[26] [27] [28] to further understand the cohomology 𝐻*(ℱ 𝑙𝑒) and 𝐻𝑇*∨(ℱ 𝑙𝑒). Using
this we will relate these cohomologies to the center of the small quantum group. Note that ℱ 𝑙𝑒 indeed has an action of 𝑇∨(𝒦) and this induces an action on 𝐻*(ℱ 𝑙𝑒)
that factors through Λ ∼= 𝑇∨(𝒦)/𝑇∨(𝒪), as connected groups act trivially on coho-mology.
We now introduce some of the algebras that will be relevant. Let 𝑢𝑞 := 𝑢𝑞(𝐺) be
the small quantum group, 𝑈𝑞 := 𝑈𝑞(𝐺) the Lusztig quantum group and U𝑞:= U𝑞(𝐺)
the Kac De Concini quantum group associated to the group 𝐺. The constructions of these algebras are given by work of Lusztig [37], De Concini, Kac and Procesi [20] [21]
[22], we will also briefly consider the corresponding algebras for certain Levi’s and Parabolic subgroups of 𝐺, but we will label them explicitly when doing so. These algebras are indeed Hopf algebras and so we can consider tensor products of repre-sentations.
Here we briefly describe these algebras. The quantum group is an algebra over C(𝑞) generated by 𝐸𝑖, 𝐹𝑖, 𝐾𝜆, for 𝑖 running through the simple roots 𝛼𝑖 and 𝜆 ∈ Λ*. The
algebra is defined subject to the relations
𝐾0 = 1 𝐾𝜆𝐾𝜇 = 𝐾𝜆+𝜇 𝐾𝜆𝐸𝑖𝐾−𝜆 = 𝑞<𝜆,𝛼𝑖>𝐸𝑖 𝐾𝜆𝐹𝑖𝐾−𝜆 = 𝑞−<𝜆,𝛼𝑖>𝐹𝑖 [𝐸𝑖, 𝐹𝑗] = 𝐾𝑖− 𝐾𝑖−1 𝑞𝑑𝑖 − 𝑞−𝑑𝑖
Here 𝐾𝑖 := 𝐾𝑑𝑖𝛼∨𝑖 and 𝑑𝑖 := (𝛼𝑖, 𝛼𝑖) gives the relative lengths of the roots, where the
shortest roots have 𝑑𝑖 = 1.
Now the Kac-De Concini quantum group is the subalgebra over C[𝑞±1] generated by 𝐸𝑖, 𝐹𝑖 and 𝐾𝜆.
The Lusztig quantum group is then the algebra over C[𝑞±1] generated by the following elements: 𝐸𝑖(𝑙) := 𝐸𝑖𝑙 [𝑙]𝑑𝑖!, 𝐹 (𝑙) 𝑖 := 𝐹𝑙 𝑖 [𝑙]𝑑𝑖! and [ (︀𝐾𝜆,𝑚 𝑛 )︀]𝑑𝑖. Here [𝑙]𝑑! := ∏︀𝑙 𝑖=1 𝑞𝑖𝑑−𝑞−𝑖𝑑 𝑞𝑑−𝑞−𝑑 . The last
element is defined by Lusztig and is a quantum deformation of the choose function (︀𝑥−𝑚
𝑛 )︀.
We can specialize both these algebras at 𝑞 an 𝑙th root of unity for 𝑙 an odd prime, we will denote the algebra 𝑈q, respectively Uq, when considering it as an algebra over
C[𝑞±1] and 𝑈𝑞, respectively U𝑞, when considering the specialization at an 𝑙th root of
unity.
Note that by the definition of the algebras we have a natural map U𝑞 → 𝑈𝑞. For the
above specialization the image is precisely the small quantum group 𝑢𝑞.
we denote by 𝑍𝑙. This algebra is generated by 𝐾𝜆𝑙, and 𝐸𝑖𝑙, 𝐹𝑖𝑙 for the simple roots
and analogous elements for all other positive roots. We have an isomorphism 𝑍𝑙 ∼=
C[𝐵−𝐵], the algebra of functions of the open cell of the group 𝐺. We can then also
define the small quantum group as 𝑢𝑞 ∼= U𝑞⊗𝑍𝑙C𝑖𝑑, the quotient of the Kac-de Concini
quantum group with the 𝑙-center specialized at the identity.
For a weight 𝜆 of 𝑇 , we can define a highest weight baby Verma 𝑍(𝜆) := 𝑢𝑞⊗𝑢𝑞(𝐵)C𝜆,
by induction from the Borel. Here the action of 𝑢𝑞(𝐵), factors through the map
𝑢𝑞(𝐵) → 𝑢𝑞(𝑇 ) and the action of this is determined by the weight 𝜆.
We also have the quantum Frobenius 𝑈𝑞 → ˆ𝒰 (g) as introduced by Lusztig [37].
Here the target is a completion of the enveloping algebra with finite dimensional representations given by 𝑅𝑒𝑝(𝐺). Thus under this map we can consider a functor 𝑅𝑒𝑝(𝐺) → 𝑈𝑞− 𝑚𝑜𝑑. The map is determined by the property that 𝐸𝑖 ↦→ 0, 𝐹𝑖 ↦→ 0
and 𝐸(𝑙) ↦→ 𝑒𝑖, 𝐹𝑖(𝑙)↦→ 𝑓𝑖, where 𝑒𝑖 and 𝑓𝑖 are the generators of the enveloping algebra.
Note that this map acts like the counit on the small quantum group. In fact it can be considered as a Hopf algebra quotient by the normal subalgebra given by 𝑢𝑞.
The above are Hopf algebras and so they have an adjoint action on itself. Then we know:
𝑍(𝑈𝑞) : = (𝑈𝑞)𝑈𝑞
𝑍(𝑢𝑞) : = 𝑢𝑢𝑞𝑞
ie the fixed points under the adjoint representation are exactly given by the center. Further under the adjoint action of 𝑈𝑞 on itself 𝑢𝑞 is stable. This is the condition for
a Hopf subalgebra to be normal. Thus we can consider first taking 𝑢𝑞 fixed points
and then we will have a remaining action by the quotient, ie we have an induced 𝐺 representation by the above remarks. It thus follows that we have a 𝐺 action on 𝑍(𝑢𝑞).
Further note that from the above it follows that 𝑍(𝑈𝑞) ∩ 𝑢𝑞 = 𝑍(𝑢𝑞)𝐺.
For the identifications needed in the following statements we will need an isomorphism ¯
would be to consider the above quantum groups over ¯Q𝑙′ instead, which is equivalent,
but introduces complication into the notation.
2.1.2
Overview of results
The main result of this chapter is given by the following commutative diagram
𝐻*(ℱ 𝑙) 𝑍(𝑢^0 𝑞− 𝑚𝑜𝑑)𝐺 𝐻*(ℱ 𝑙𝑒)Λ 𝑍(𝑢 ^ 0 𝑞− 𝑚𝑜𝑑)𝑇
Here the RHS of the diagram is given by the 𝑇 , respectively 𝐺 fixed points of the action on 𝑍(𝑢𝑞− 𝑚𝑜𝑑^0) the center of a regular block of 𝑢𝑞 and the 𝐺 action is given
as described above.
In this chapter we introduce several maps from several cohomologies to the centers we are interested in. For the above maps, we know the bottom map is injective. We will also give a deformation theory setting towards a proof of the surjectivity of the top map and we will show that indeed all the central elements deform in the first order infinitesimal deformation, but it remains an open conjecture to continue lifting these central elements to higher order infinitesimal deformations and get the required surjectivity.
In upcoming work with Losev together with some remarks of Shan and Vasserot it can be shown that the map of cohomologies is surjective in Type A.
It remains a conjecture that the bottom map is surjective giving an isomorphism 𝐻*(ℱ 𝑙𝑒)Λ ∼= 𝑍(𝑢𝑞)𝑇. Using this and the above remarks in Type A it would follow
that the 𝐺 action on the center 𝑍(𝑢^0
𝑞− 𝑚𝑜𝑑) is indeed trivial.
2.1.3
Outline of the Chapter
In Sections 2-5 we will construct the map to the center from 𝐻*(ℱ 𝑙𝑒). To do this we
will require to understand some ideas of 𝑇∨-equivariant cohomology as worked out by Goresky, Kottwitz and MacPherson [26] [27] [28], which will be stated in Section 2.
In Section 3 and 4 we study the representation theory of 𝑢𝑞 and certain deformations
required in the proof. Here the approach is between general results in Section 3 and particular results of 𝑆𝐿2 in Section 4. In Section 5 we put all these results together
to construct the action.
In Section 6 the map from 𝐻*(ℱ 𝑙) is constructed. Here we use the work of Arkhipov, Bezrukavnikov and Ginzburg [4], as well as the work of Bezrukavnikov and Yun [16], to construct the desired action.
In Section 7-11 we introduce a certain deformation theory setting, which we hope to use the surjectivity of the map 𝐻*(ℱ 𝑙) → 𝑍(𝑢^0𝑞 − 𝑚𝑜𝑑)𝐺 in a future paper. So far
we are able to prove lifting to a first order deformation using this deformation theory setting. Section 7 is devoted to just introduce the setting and certain cetegories related to it. In Section 8 we determine the center of the flat deformation category. We then check the compatibility between this new map to the center and the map induced from 𝐻*(ℱ 𝑙) in Section 9. We prove the first order deformation we are so far able to prove in Section 10 and 11. To do this we use Section 10 to determine the vanishing of a certain Hochschild cohomology group and in Section 11 we use the vanishing of this group to prove the lifting of central elements to first order.
In Section 12 we give an algebraic interpretation of the action of 𝐻*(ℱ 𝑙). This interpretation is interesting in its own right, but is also necessary for the following section.
In Section 13 we check the compatibility of both actions determined before and thus prove the commutativity of the above diagram.
2.2
Equivariant cohomology and GKM
In this section we introduce results by Goresky, Kottwitz and MacPherson [26], giving an explicit description of 𝑇 -equivariant cohomology under some conditions. This result will be needed to give an explicit description of 𝐻𝑇*∨(ℱ 𝑙𝑒) and use this to
identify it with the center of a certain category.
Consider an algebraic variety 𝑋 with an action of a torus 𝑇 . Recall that the definition of equivariant cohomology is given by
𝐻𝑇*(𝑋) = 𝐻*(𝑋 ×𝑇 𝐸𝑇 )
ie the cohomology of the Borel construction, where here 𝐸𝑇 → 𝐵𝑇 is the universal 𝑇 -bundle.
We further know that 𝐻𝑇*(𝑝𝑡) = C[t].
Now assume that 𝑋 has finitely many 𝑇 fixed points. Then we have the following result
Lemma 1. Let 𝑋 be an algebraic variety with an action of a torus 𝑇 . Further assume that 𝑋 has finitely many fixed points 𝐹 = 𝑋𝑇. Then we have the map
𝐻𝑇*(𝑋) → 𝐻𝑇*(𝑋𝑇) = ⊕𝐹C[t] given by restricting to fixed points, is injective.
Further from the above description of the equivariant cohomology, we have a spectral sequence
𝐻𝑝(𝐵𝑇, 𝐻𝑞(𝑋)) ⇒ 𝐻𝑇𝑝+𝑞(𝑋).
An algebraic variety 𝑋 is called equivariantly formal, if the above spectral sequence collapses in the 𝐸2-page.
Consider the above setting for 𝑋 and a 1-dimensional orbit 𝐸 of 𝑇 . Then the closure of 𝐸 contains 2 fixed points 𝑥0, 𝑥∞ and the action of 𝑇 on 𝐸 factors through some
character 𝜒 : 𝑇 → G𝑚.
Under these conditions we have the following Proposition
Proposition 1. Let 𝑋 be an algebraic variety with a torus 𝑇 action, such that 𝑋 is equivariantly formal, has finitely many fixed points and 1-dimensional orbits, then we have the restriction map
is injective and further the image is given by tuples (𝑓𝑥)𝑥∈𝐹, such that for every
1-dimensional orbit whose closure contains 𝑥0 and 𝑥∞ and with character 𝜒 we have
𝑓𝑥0 − 𝑓𝑥∞ ∈ (𝜒)
where we consider 𝜒 ∈ C[t].
We can also consider a similar result in the case of ind-schemes by taking limits. The conditions of finitely many fixed points and 1-dimensional orbits becomes the condition of having isolated fixed points and for any pair of fixed points only finitely many 1-dimensional orbits connecting them.
2.3
Basics of the representation theory of the small
quantum group
In this section we will introduce some basic results of the representation theory of the small quantum group 𝑢𝑞 and related algebras that we will use in the following
section. The results here can be found in Jantzen’s work [31].
We introduce the algebra U𝑡,𝑞 := U𝑞 ⊗𝑍𝑙𝒪0, where 𝒪0 is the algebra of the formal
neighbourhood of the identity considered as an point in the maximal torus 𝑇 ⊂ 𝐵−𝐵.
Note that 𝒪0 is isomorphic to the algebra of the formal neighbourhood of 0 in t, the
Lie algebra of the torus 𝑇 .
Further we can see that all the algebras we consider have a grading by the root lattice. We can consider the categories of weight graded representations 𝑢𝑞 − 𝑚𝑜𝑑𝑔𝑟
and U𝑡,𝑞 − 𝑚𝑜𝑑𝑔𝑟, ie graded representations of the corresponding algebras with the
action compatible with the grading of the algebras and with compatible action of the torus subalgebras with the grading. This can be understood as the representation category of a Harish-Chandra pair (𝑢𝑞, 𝑇 ) and (U𝑡,𝑞, 𝑇 ) respectively.
The simple modules of U𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟 are supported at the unique closed point of the
formal neighbourhood algebra 𝒪0 of 𝑇 at the identity, hence the finitely generated
The simple modules 𝐿𝜆 for 𝑢𝑞− 𝑚𝑜𝑑𝑔𝑟 are classified by their highest weight 𝜆 ∈ Λ.
Further there are baby Verma modules 𝑍(𝜆) of highest weight 𝜆, which are obtained by induction from the small quantum group of the positive Borel part. Consider also the projective covers 𝑃 (𝜆) of the simple modules 𝐿𝜆.
To state the following result we need to introduce a certain action of ̃︁𝑊 on Λ. Consider the usual action of ̃︁𝑊 on Λ. We can now introduce the ∙-action given by stretching the action of translations by 𝑙 and shifting the center of the action to −𝜌. We have then the following result known as the Linkage principle
Lemma 2. If 𝐸𝑥𝑡1(𝐿𝜆, 𝐿𝜇) ̸= 0 then ∃𝑤 ∈ ̃︁𝑊 such that 𝑤 ∙ 𝜆 = 𝜇
Thus we can consider the category U^0
𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟 given by the Serre subcategory
containing simples of highest weights 𝑤 ∙ 0 and similarly for 𝑢^0𝑞− 𝑚𝑜𝑑𝑔𝑟 and 𝑢^0
𝑞− 𝑚𝑜𝑑.
We can clearly see that the simple modules are then in bijection with ̃︁𝑊 .
By work of Andersen, Jantzen and Soergel [1] there are flat deformations over 𝒪0
of the projective covers 𝑃 (𝜆) of 𝐿𝜆 as 𝑢𝑞-modules. There are also flat deformations
of the baby Vermas, which are just given by the same induction construction. We denote these by ˜𝑃 (𝜆) and ˜𝑍(𝜆), respectively.
Further we will need to understand these algebras at the generic point of t and at codimension one points.
In general consider a point 𝜒 of the formal neighbourhood algebra 𝒪0. Consider 𝐿
the Levi that stabilizes 𝜒 and let 𝑃 be a parabolic with Levi 𝐿. With this setting we have the following equivalence
𝒪𝜒⊗ U𝑡,𝑞(𝐿) − 𝑚𝑜𝑑𝑔𝑟 → 𝒪𝜒⊗ U𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟
𝑀 ↦→ U𝑡,𝑞⊗U𝑡,𝑞(𝑃 )𝑀
Here we consider the action of U𝑡,𝑞(𝑃 ) by considering the action through the map
U𝑡,𝑞(𝑃 ) → U𝑡,𝑞(𝐿) and 𝒪𝜒 is the localization of 𝒪0 at the point 𝜒. Further the RHS
category is given by the central torus of 𝐿 acting semisimply.
We will use this interpretation of the category at specific points to reduce the com-putation, to the generic point and codimension one points. The relevant stabilizers
will be either 𝑇 the maximal torus, or some Levi 𝐿 of semisimple rank 1. We will thus need to study the representation theory of these algebras.
2.4
The representation theory for sl
2In this section we further explore the representation theory of the above algebras in the case of sl2. We will need this in the computation of the center below.
For U^0𝑡,𝑞(sl2) − 𝑚𝑜𝑑𝑔𝑟, recall from the above that the simples are classified by ̃︁𝑊 .
We will now state several facts about the representation theory of this algebra, that can be found in Andersen, Jantzen and Soergel (Lemma 8.10) [1].
Let 𝑠 and 𝑠0 be the simple reflections and assume WLOG 𝑠 is such that 𝑤𝑠 ∙ 0 < 𝑤 ∙ 0,
we have the deformation of the projective cover ˜𝑃 (𝑤 ∙ 0) of 𝐿𝑤∙0fits into a short exact
sequence
0 → ˜𝑍(𝑤𝑠 ∙ 0)−→ ˜𝑓 𝑃 (𝑤 ∙ 0)−→ ˜𝑔 𝑍(𝑤 ∙ 0) → 0.
Further 𝐻𝑜𝑚( ˜𝑃 (𝑤 ∙ 0), ˜𝑍(𝑤 ∙ 0)) and 𝐻𝑜𝑚( ˜𝑍(𝑤𝑠 ∙ 0), ˜𝑃 (𝑤 ∙ 0)) are both free of rank 1 spanned by the above maps.
Also 𝐻𝑜𝑚( ˜𝑍(𝑤 ∙ 0), ˜𝑃 (𝑤 ∙ 0)) and 𝐻𝑜𝑚( ˜𝑃 (𝑤 ∙ 0), ˜𝑍(𝑤𝑠 ∙ 0)) are free of rank one and let the basis be given by 𝑔′ and 𝑓′ respectively. Then 𝑓′∘ 𝑓 = ℎ𝑢 and 𝑔 ∘ 𝑔′ = ℎ𝑣 for
ℎ the variable corresponding to the root 𝛼 and 𝑢 and 𝑣 are units in 𝒪𝛼, where this
is the local algebra corresponding to 𝛼, where we consider this as a codimension 1 point in 𝒪0.
Now we will try to compute the algebra 𝐸𝑛𝑑(⊕𝑤∈̃︁𝑊𝑃 (𝑤 ∙ 0)), which governs this˜ category. Note first that if you invert ℎ, the category becomes semisimple with simples given by C(ℎ) ⊗ ˜𝑍(𝜆). This follows by considering the above result at the generic point and by understanding that the category corresponding to the torus 𝑇 is indeed semisimple. It thus follows from the above short exact sequences that this is a subalgebra of ∏︀
̃︁
𝑊 𝑀 𝑎𝑡2(C(ℎ)). We then get the following result.
This algebra is given by (maybe infinite) linear combinations over C[[ℎ]] of the fol-lowing elements in the above algebra ∏︀
̃︁ 𝑊 𝑀 𝑎𝑡2(C((ℎ))): (. . . , 0, ⎡ ⎣ 0 0 1 0 ⎤ ⎦, 0, . . . ) (2.1) (. . . , 0, ⎡ ⎣ 0 ℎ 0 0 ⎤ ⎦, 0, . . . ) (2.2) (. . . , 0, ⎡ ⎣ ℎ 0 0 0 ⎤ ⎦, 0, . . . ) (2.3) (. . . , ⎡ ⎣ 0 0 0 1 ⎤ ⎦, ⎡ ⎣ 1 0 0 0 ⎤ ⎦, 0, . . . ) (2.4)
Proof. Note that we can use 𝑓 and 𝑔 given above to identify ˜𝑃 (𝑤 ∙ 0) as the direct sum of two baby Vermas as required. Using these we can define a map ˜𝑃 (𝑤 ∙ 0) →
˜
𝑍(𝑤 ∙ 0) → ˜𝑃 (𝑤𝑠0 ∙ 0). This will give exactly the first element.
We can use the 𝑓′ and 𝑔′ similarly to define a map ˜𝑃 (𝑤 ∙ 0) → ˜𝑃 (𝑤𝑠0 ∙ 0). Since
we use 𝑓 and 𝑔 to define the embedding and using the properties 𝑓′ ∘ 𝑓 = ℎ𝑢 and 𝑔 ∘ 𝑔′ = ℎ𝑣, we can get the second up to a unit.
The third element is just given by composing the first two and the fourth is just the idempotent corresponding to ˜𝑃 (𝑤 ∙ 0).
Further note that by the conditions of 𝑓 , 𝑔, 𝑓′and 𝑔′being a basis, we get that the first two maps are indeed bases for 𝐻𝑜𝑚( ˜𝑃 (𝑤∙0), ˜𝑃 (𝑤𝑠∙0)) and 𝐻𝑜𝑚( ˜𝑃 (𝑤𝑠∙0), ˜𝑃 (𝑤∙0)). Note further that as only these projectives have a common baby Verma subquotients there are no other non-zero homomorphisms between distinct indecomposable pro-jectives.
It remains to show that the endomorphisms of ˜𝑃 (𝑤 ∙ 0) are spanned by the last two. If we have any endomorphism of ˜𝑃 (𝑤 ∙ 0), we can assume, by taking away some multiple of the identity, that the image lies in the image of ˜𝑍(𝑤𝑠 ∙ 0) → ˜𝑃 (𝑤 ∙ 0). Thus we get by the description of such homomorphisms that this is just a multiple of the third map. The result now follows as required.
Now we will use the above description of the algebra governing the category, to determine the center.
Lemma 4. The center of the algebra 𝐸𝑛𝑑(⊕𝑤∈̃︁𝑊𝑃 (𝑤 ∙ 0)) given by the above descrip-˜ tion is given by elements
(. . . , ⎡ ⎣ 𝐴𝑤𝑠 0 0 𝐴𝑤𝑠 ⎤ ⎦, ⎡ ⎣ 𝐴𝑤 0 0 𝐴𝑤 ⎤ ⎦, . . . )
such that ℎ|(𝐴𝑤− 𝐴𝑤𝑠) and ℎ|(𝐴𝑤 − 𝐴𝑤𝑠0)
Proof. Assume we have a central element given by
(. . . , ⎡ ⎣ 𝐴𝑤𝑠 𝐵𝑤𝑠 𝐶𝑤𝑠 𝐷𝑤𝑠 ⎤ ⎦, ⎡ ⎣ 𝐴𝑤 𝐵𝑤 𝐶𝑤 𝐷𝑤 ⎤ ⎦, . . . )
Then by commuting with the element (1) and (2), in the previous lemma we get 𝐵𝑤 = 𝐶𝑤 = 0 and 𝐴𝑤 = 𝐷𝑤.
But note that the element has to be spanned by the elements in the previous lemma, thus we get that 𝐴𝑤− 𝐴𝑤𝑠 and 𝐴𝑤 − 𝐴𝑤𝑠0 are divisible by ℎ.
Note that an element satisfying these conditions does indeed lie in the endomorphisms and it is clearly central as required.
We will use this description to compare conditions for central elements, with the conditions given by the GKM description of the cohomology of ℱ 𝑙𝑒.
2.5
The map from 𝐻
*(ℱ 𝑙
𝑒)
In this section we will find the center of the above deformation of the small quantum group 𝑢𝑞 and use this to construct central elements in 𝑢𝑞.
We will identify the center of the category U^0𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟 explicitly. To do this we will
need the description of 𝐻𝑇*∨(ℱ 𝑙𝑒).
apply the above results.
From the previous section to get a description of this cohomology we need to give a description of fixed points and 1-dimensional orbits. From GKM [28] it follows that the fixed points are given by ̃︁𝑊 the affine Weyl group and the 1-dimensional orbits join the points 𝑥 and 𝑥𝑠𝛼,𝜖, for 𝜖 ∈ {0, 1} and 𝛼 a positive root, where 𝑠𝛼,𝑛
is the affine reflection on the affine hyperplane given by the condition < 𝜆, 𝛼 >= 𝑛. The character of this orbit is given by ¯𝑥(𝛼), where ¯𝑥 ∈ 𝑊 is the image of 𝑥 in the finite Weyl group 𝑊 under the quotient map ̃︁𝑊 → 𝑊 . This set of fixed points and 1-dimensional orbits satisfy the conditions for the GKM description, so we can use the above to give a description of 𝐻𝑇*∨(ℱ 𝑙𝑒).
To state the Lemma correctly we will need to consider a slight modification on the cohomology. To be precise we need to consider ˆ𝐻*(𝑋) := ∏︀ 𝐻𝑖(𝑋), instead of
𝐻*(𝑋) := ⊕𝐻𝑖(𝑋). In the case of the 𝑇 -equivariant cohomology, we get that ˆ𝐻𝑇*(𝑋) is a module over ˆ𝐻*(𝑝𝑡) ∼= C[[t]] ∼= 𝒪0. The same conditions for the GKM description
applies in this setting.
Lemma 5. We have the following isomorphism
𝑍(U^0𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟) ∼
= ˆ𝐻𝑇*∨(ℱ 𝑙𝑒)
Proof. There are several ways of understanding the construction of these central ele-ments from 𝑢𝑞− 𝑚𝑜𝑑𝑔𝑟. We can either see this from a coherent realization using work
by Bezrukavnikov [9] or we can use this using work by Andersen, Jantzen and Soergel [1]. They are really the same proof, but two ways of describing the necessary object. We start by giving the proof using the work of Andersen, Jantzen and Soergel. First note that the finitely generated simples are supported on the unique closed point of the completion of t at 0 and thus we have the simples are given by the simples of 𝑢^0
𝑞. These are in bijection with ̃︁𝑊 under the map 𝑤 ↦→ 𝐿𝑤∙0, where here the ∙ action
is the usual action, but stretched by 𝑙 and centered at −𝜌. Further by Andersen, Jantzen and Soergel [1], there are flat deformations over t of the projective covers of 𝐿𝜆 as a 𝑢𝑞-module.
Thus we get that the center of U^0𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟 is torsion free over t.
So we have
𝑍(U^0𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟
) ⊂ 𝑍(C((t)) ⊗ U^0𝑡,𝑞− 𝑚𝑜𝑑 𝑔𝑟)
and further the subset is given by understanding the category at codimension 1 points of t𝑡.
Now note that understanding the category at the generic point, is understanding representation at a generic semisimple character. Recall that from the above we have the following equivalence of categories
C((t)) ⊗ U𝑡,𝑞(𝑇 ) − 𝑚𝑜𝑑𝑔𝑟 ∼= C((t)) ⊗ U𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟
Further the category on the LHS is clearly semisimple, as it is given by a commutative algebra and each character has no self-extensions.
In particular we get the center of the category at the generic point is given by a copy of C((t)) for each simple object. Thus we get
𝑍(U^0𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟) ⊂ ∏︁
𝑤∈̃︁𝑊
C((t)).
We can further look at the specialization of the category at the different codimension 1 points. For each codimension 1 point the action of the center has to preserve the corresponding baby Verma, so we get immediately that
𝑍(U^0𝑡,𝑞− 𝑚𝑜𝑑𝑔𝑟) ⊂ ∏︁
𝑤∈̃︁𝑊
C[[𝑡]].
Further note that for a generic codimension 1 point 𝜒 the Levi that fixes 𝜒 is given by 𝑇 , thus we get these categories are semisimple as above.
The only codimension 1 characters where something interesting happens is at roots 𝛼. For these we have the Levi that fixes this is generated by 𝑇 and the one dimensional root subgroups corresponding to ±𝛼.
From the previous section we thus get the condition at the root 𝛼 is given by the relation that an element (𝑓𝑤) ∈
∏︀
𝑤∈̃︁𝑊C[[𝑡]] to preserve the 𝛼 specialized category,
we need that 𝑓𝑤 − 𝑓𝑤𝑠𝛼,𝜖 is divisible by 𝛼 for 𝜖 ∈ {0, 1}. This follows immediately
from the description in the case of sl2 as stated above.
Note that these are exactly the same conditions as those given by the 1 dimensional orbits in ℱ 𝑙𝑒 and thus by the GKM-description of the center we get
ˆ 𝐻𝑇*∨(ℱ 𝑙𝑒) ∼= 𝑍(U ^ 0 𝑡,𝑞− 𝑚𝑜𝑑 𝑔𝑟) as required.
For the second approach, we will note that the representation of 𝑢^0
𝑞 can be given an
interpretation as coherent sheaves on ˜g×𝐿
g {0} and further
𝐷(𝑢^0𝑞− 𝑚𝑜𝑑𝑔𝑟) ∼
= 𝐷𝐺𝐶𝑜ℎ(𝑇 ∖˜g×𝐿 g {0})
Thus we can have a flat deformation over t, given by 𝐷𝐶𝑜ℎ(𝑇 ∖˜g×𝐿 g t).
We can again reduce the question to understanding the center at the generic point of t and codimension one points. Then specializing at the generic point of t, the components of ˜g×𝐿
g t are in bijection with 𝑊 corresponding to the 𝑇 -fixed points of
𝐺/𝐵. We can thus consider the structure sheaf 𝒵𝑡𝜆𝑤for the corresponding component
given by 𝑤 ∈ 𝑊 and twist it by a character 𝜆 of 𝑇 .
These objects play the role of the flat deformations of the baby Verma and we can check that 𝐸𝑛𝑑(𝒵𝑡𝜆𝑤) ∼= C[t].
This gives the inclusion map into the product of C[t] over ̃︁𝑊 .
Now to compute the isomorphism, we reduce it to 𝑆𝐿2 and thus we can reduce it to
understanding 𝐷𝐶𝑜ℎ(𝑇 ∖˜g×𝐿
g t) in the case of 𝑆𝐿2.
This now reduces to a computation similar to the above for 𝑆𝐿2, but in this geometric
context. We will skip the details of this approach.
Corollary 1. We have the following injective maps
𝑍(𝑢^0𝑞− 𝑚𝑜𝑑𝑔𝑟) ←˒ 𝐻*
(ℱ 𝑙𝑒)
𝑍(𝑢^0𝑞− 𝑚𝑜𝑑)𝑇 ←˒ 𝐻*
(ℱ 𝑙𝑒)Λ
Proof. The category 𝑢^0
𝑞 − 𝑚𝑜𝑑𝑔𝑟 is the full subcategory of U ^ 0
𝑡,𝑞 − 𝑚𝑜𝑑𝑔𝑟 of modules
for which C[[t]] acts trivially. The central C[[t]] action corresponds to multiplication by 𝐻𝑇*(𝑝𝑡) induced by the obvious map to 𝐻𝑇*(ℱ 𝑙𝑒). This is clear by acting on the
deformed Vermas. Thus we get the first injective map after quotienting by this action. The last thing to note is that
𝐻𝑇*∨(ℱ 𝑙𝑒) ⊗𝐻* 𝑇 ∨(𝑝𝑡)C
∼
= 𝐻*(ℱ 𝑙𝑒)
using that ℱ 𝑙𝑒 is equivariantly formal.
The same is true for the completed cohomology and since ℱ 𝑙𝑒 is finite dimensional,
we have that ˆ𝐻*(ℱ 𝑙𝑒) ∼= 𝐻*(ℱ 𝑙𝑒).
These central elements can also be constructed using the coherent construction de-scribed above.
To see the second result, note that the forgetful functor 𝑢^0𝑞 − 𝑚𝑜𝑑𝑔𝑟 → 𝑢^0
𝑞− 𝑚𝑜𝑑 is
a degrading functor, with the translation functor being given by tensoring with the 1-dimensional representations with weight 𝑝𝜆 for some 𝜆 ∈ Λ.
Note that the action of Λ on the center is just given by translating the 𝑍(𝑤 ∙ 0) to 𝑍(𝑡𝜆𝑤 ∙ 0). Thus under the isomorphism with ˆ𝐻𝑇*∨(ℱ 𝑙𝑒), this action corresponds to
the action induced by the translation action of Λ on ℱ 𝑙𝑒.
Now note that central elements invariant under the translation functors descend to the degraded category. Thus we get the induced map to the center. Now recall that we can consider the graded category as the category of the Harish-Chandra pair (𝑢𝑞, 𝑇 ). Thus central elements of 𝑢𝑞 that remain central in the Harish-Chandra pair
2.6
Map from 𝐻
*(ℱ 𝑙)
In this section we introduce some categories that we will need going forward and we will introduce some maps of certain cohomologies to the center.
In this section we use results orginally proven in the Lusztig program by combining work of Kazdhan, Lusztig [35] and Kashiwara, Tanisaki [33]. This was also proven in Arkhipov, Bezrukavnikov, Ginzburg [4]. We will also use work of Arkhipov, Bezrukan-vikov [2] and Bezrukavnikov, Yun [16].
We start by stating a few Theorems we will use
Theorem 1. (KT[33]+KL[34], ABG[4]) We have an equivalence of categories
𝑃 𝑒𝑟𝑣𝐼0(𝒢𝑟) ∼= 𝑈
^ 0
𝑞 − 𝑚𝑜𝑑
Theorem 2. (BY[16]) We have a Koszul duality equivalence of categories
𝐷𝐼0,𝑚(𝒢𝑟) ∼= 𝐷𝐼𝑊,𝑚(ℱ 𝑙)
of mixed sheaves in each case. Here we have the twist functor (1) on the LHS is sent to the functor (1)[1] on the RHS
We then get the following:
Lemma 6. We have a map 𝐻*(ℱ 𝑙) → 𝑍(𝑈^0
𝑞 − 𝑚𝑜𝑑)
Proof. First we know that 𝐷𝐼𝑊(ℱ 𝑙) ⊂ 𝐷(ℱ 𝑙) is a full subcategory.
Using this note that the cohomology of 𝐻*(ℱ 𝑙) acts on ℱ ∈ 𝐷(ℱ 𝑙), by giving maps ℱ ∼= ℱ ⊗ Cℱ 𝑙 → ℱ ⊗ Cℱ 𝑙[𝑖] ∼= ℱ [𝑖], where we consider 𝑥 ∈ 𝐻𝑖(ℱ 𝑙) as a map
Cℱ 𝑙 → Cℱ 𝑙[𝑖].
Further we can consider similarly the action in the mixed category by considering the weight of a cohomological element. Note that since the cohomology of ℱ 𝑙 is pure we have the maps are given by Cℱ 𝑙 → Cℱ 𝑙(𝑖)[𝑖].
Thus under the Koszul duality equivalence of BY we get induced natural maps ℱ → ℱ (𝑖), for ℱ ∈ 𝐷𝐼0,𝑚(𝒢𝑟).
Thus after forgeting the mixed structure we get natural endomorphism of the identity functor and hence we get a map 𝐻*(ℱ 𝑙) → 𝑍(𝑃 𝑒𝑟𝑣𝐼0(𝒢𝑟)).
Now by considering the equivalence in ABG we get immediately the required result.
Now we can consider a further lemma
Lemma 7. The above map
𝐻*(ℱ 𝑙) → 𝑍(𝑈𝑞^0− 𝑚𝑜𝑑) lands in the small quantum group 𝑢𝑞∩ 𝑍(𝑈
^ 0 𝑞 − 𝑚𝑜𝑑) = 𝑍(𝑢 ^ 0 𝑞− 𝑚𝑜𝑑)𝐺.
To prove this lemma, we will need a further result. To state this result note that 𝑈^0
𝑞 − 𝑚𝑜𝑑 has an action from 𝑅𝑒𝑝(𝐺). To see this recall that we have the quantum
Frobenius map, a map of Hopf algebras, as described by Lusztig. This gives us a functor 𝑅𝑒𝑝(𝐺) → 𝑈𝑞− 𝑚𝑜𝑑 under pullback of the quantum Frobenius. Thus as this
is a map of tensor categories, we have an action of 𝑅𝑒𝑝(𝐺) on 𝑈𝑞− 𝑚𝑜𝑑 and further
this does not change the central character and so we have an action as described. Using this note that we can consider the algebra in 𝑅𝑒𝑝(𝐺) given by 𝒪𝐺, the algebra
of regular function on 𝐺, with the action given by left translations. We can thus consider 𝒪𝐺-modules on the category 𝑈𝑞^0− 𝑚𝑜𝑑 and we can state the following result
by Arkhipov and Gaitsgory [5].
Proposition 2. (AG[5]) We have an equivalence of categories
𝒪𝐺− 𝑚𝑜𝑑(𝑈 ^ 0 𝑞 − 𝑚𝑜𝑑) ∼= 𝑢 ^ 0 𝑞− 𝑚𝑜𝑑
Now we prove the Lemma using this result.
Proof. It follows from the above results that a central element of 𝑈𝑞^0− 𝑚𝑜𝑑 lies in 𝑢𝑞
if and only if the 𝒪𝐺 algebra action is preserved.
equivalent for a central element to satisfy the following maps
𝑧ℱ ⊗ 𝑖𝑑𝑉, 𝑧ℱ ⊗𝑉 : ℱ ⊗ 𝑉 → ℱ ⊗ 𝑉
are equal for every ℱ ∈ 𝑈𝑞− 𝑚𝑜𝑑 ^
0 and 𝑉 ∈ 𝑅𝑒𝑝(𝐺).
To check this we need to understand the above central action on objects of the form ℱ ⊗ 𝑉 . Note that the action is defined on the Koszul dual, so we will prove this result on that category.
Now note that on the Koszul dual side, we have the action of 𝑅𝑒𝑝(𝐺) is given by convolution with the objects 𝑍(𝑉 ) as described by Gaitsgory [24].
Further note that as stated in Arkhipov, Bezrukavnikov [2], these objects are tilting when considered as objects of 𝑃 𝑒𝑟𝑣𝐼𝑊(ℱ 𝑙) by applying 𝐴𝑣𝐼𝑊.
It follows that 𝐸𝑥𝑡*(𝑍(𝑉 ), 𝑍(𝑉 )) = 0 for * > 0. Thus as 𝐻*(ℱ 𝑙) acts through higher order self-extension, we see that the action on 𝑍(𝑉 ) are trivial.
Further to understand the action on convolution products we consider the definition of this product.
Consider the convolution space 𝐶𝑜𝑛𝑣 = 𝐺∨(𝒦) ×𝐼𝐺∨(𝒦)/𝐼, then we get the
convo-lution under the following maps
𝐶𝑜𝑛𝑣
ℱ 𝑙 ℱ 𝑙 𝐼∖ℱ 𝑙
𝑝1
𝑚 𝑝2
Thus to define the convolution we need to consider 𝑚*(𝑝1× 𝑝2)*(ℱ 𝑍(𝑉 )).
From this definition we see the action on the convolution is going to be through the map 𝐻*(ℱ 𝑙) → 𝐻*(𝐶𝑜𝑛𝑣) given by the map 𝑚.
Now note that using the typical argument for equivariantly formal relation between usual cohomology and equivariant cohomology, we get
𝐻*(𝐶𝑜𝑛𝑣) ∼= 𝐻*(ℱ 𝑙) ⊗𝐻𝐼*(𝑝𝑡)𝐻𝐼*(ℱ 𝑙)
Using this we note that the action on ℱ ⊗𝑍(𝑉 ) is given by the action of 𝐻*(ℱ 𝑙)⊗𝐻* 𝐼(𝑝𝑡)
𝐻𝐼*(ℱ 𝑙), respectively on each factor. But as stated above 𝑍(𝑉 ) are tilting on 𝑃 𝑒𝑟𝑣𝐼𝑊(ℱ 𝑙)
and thus the action on this is trivial, and thus we get the action is just given by the action on ℱ , ie we get that the maps 𝑧ℱ ⊗ 𝑖𝑑𝑉 and 𝑧ℱ ⊗𝑉 are indeed equal and thus
by the Proposition stated above this proves the result as required.
2.7
A deformation theory setting
In this section we introduce a different map to the center that is more amenable to deformation theory arguments. To do this we will use some results by Mircović and Riche [45] [49], Arkhipov and Bezrukavnikov [2].
Consider the Lie algebra g of the group 𝐺. With this we can consider the following stacks 𝐺∖˜g ∼= 𝐵∖b, where 𝐵 is the Borel and b its Lie algebra and ˜g = {(b′, 𝑥)|𝑥 ∈ b⊂ g, a Borel subalgebra} is the Grothendieck-Springer resolution. Similarly we can define ˜𝒩 = {(b′, 𝑥)|𝑥 ∈ [b′, b′] ⊂ g} ⊂ ˜g, the Springer resolution.
We have the following result
Theorem 3. (AB[2]) We have an equivalence of categories
𝐷𝐶𝑜ℎ(𝐺∖ ˜𝒩 ) ∼= 𝐷𝐼𝑊(ℱ 𝑙)
Further there is a linear Koszul duality as proven by Mircović and Riche which states
Theorem 4. (MR[45], R[49]) We have a Koszul duality equivalence of categories
𝐷𝐶𝑜ℎ(𝐺 × G𝑚∖ ˜𝒩 ) ∼= 𝐷𝐺𝐶𝑜ℎ(𝐺 × G𝑚∖{0} ×𝐿g ˜g).
Here we have the twist functor (1) on the LHS is sent to the functor (1)[1] on the RHS
Here we are considering the category of 𝐺-equivariant DG-coherent sheaves on the derived 0-fiber of the Springer resolution ˜g→ g.
Using these together we can give a further equivalence of categories
Theorem 5. We have an equivalence of categories
𝐷𝑏(𝑈𝑞^0− 𝑚𝑜𝑑) ∼= 𝐷𝐺𝐶𝑜ℎ(𝐺∖{0} ×𝐿g ˜g)
This gives a description of the category of interest as a category of DG-coherent sheaves on some derived algebraic stack. Further note that we have a flat deformation of this stack
˜ g×𝐿
g {0} ˓→ ˜g
In the following sections we will describe the center of this flat deformation, ie we will explicitly describe the center of 𝐷𝐶𝑜ℎ(𝐺∖˜g). We will then include a deformation theory argument to prove that the central elements of interest coming from the small quantum group lift to this flat deformation to first infinitesimal deformation and that the image of the center from the deformation is given by the same as by the above map from 𝐻*(ℱ 𝑙).
2.8
The center of the flat deformation
In this section we describe the center of the category 𝐷𝐶𝑜ℎ(𝐺∖˜g).
To do this we will give an interpretation of this as a category of Perverse sheaves.
Theorem 6. (B[9]) We have an equivalence of categories
𝐷𝐼𝑊(̃︁ℱ 𝑙) ∼= 𝐷𝐶𝑜ℎ𝒩̃︀(𝐺∖˜g)
Here the RHS is the category of complexes on 𝐺∖˜g, set theoretically supported on ̃︀𝒩
Here ̃︁ℱ 𝑙 is the 𝑇 -bundle over ℱ 𝑙 as described in Bezrukavnikov, Yun [16].
Further we have the following further Koszul duality equivalence, as proven in Bezrukavnikov, Yun [16].
Theorem 7. (BY[16]) We have a Koszul duality equivalence of categories
𝐷𝐺∨(𝒪),𝑚(ℱ 𝑙) ∼= 𝐷𝐼𝑊,𝑚(̃︁ℱ 𝑙)
of mixed sheaves in each case. Here we have the twist functor (1) on the LHS is sent to the functor (1)[1] on the RHS.
Now we will use these results to describe the center.
We will follow a general procedure that works in a more general setting.
First we recall the notion of highest weight category as introduced by Beilinson, Ginzburg and Soergel [8] and as explained in lecture notes by Losev.
Let 𝑘 be a field. A highest weight category 𝒞 is a 𝑘-linear artinian category, with a poset structure on the set of irreducibles Λ and the following structures
∙ A collection of standard objects ∆(𝜆) for 𝜆 ∈ Λ ∙ 𝐻𝑜𝑚(∆(𝜆), ∆(𝜇)) ̸= 0 ⇒ 𝜆 ≤ 𝜇
∙ 𝐸𝑛𝑑(∆(𝜆)) = 𝑘
∙ 𝒞 has enough projectives. If we denote 𝑃 (𝜆), the projective cover of 𝐿(𝜆) the simple corresponding to 𝜆 ∈ Λ. Then 𝑃 (𝜆), surjects on ∆(𝜆) and the kernel is filtered by ∆(𝜇) for 𝜇 > 𝜆.
Now we will state the general setting where we can compute the center.
Lemma 8. Let 𝒞 a highest weight category. Further assume that Λ has a minimum denoted by 0 and also assume that the socle of ∆(𝜆) is given by 𝐿(0).
Then the restriction map 𝑍(𝒞) ˓→ 𝐸𝑛𝑑(𝑃 (0)) is injective.
Proof. Note that 𝐸𝑛𝑑(⊕𝑃 (𝜆)) governs this category and so the center is a subalgebra of ∏︀ 𝐸𝑛𝑑(𝑃 (𝜆)).
Thus to prove the result we just need to check that if for a central element 𝑧, we have 𝑧 : 𝑃 (0) → 𝑃 (0) is 0, then 𝑧 = 0.
non-trivially on some 𝑃 (𝜆). Note that by assumption of the highest weight category, we know 𝑃 (𝜆) is ∆-filtered. Thus if we have a non-trivial map 𝑃 (𝜆) → 𝑃 (𝜆), we get a non-zero map 𝑃 (𝜆) → ∆(𝜇) for some 𝜇, by filtering the map. This map has 𝐿(0) the socle in the image, thus by projectivity we can lift this map to get a map 𝑓 : 𝑃 (0) → 𝑃 (𝜆), such that after composing with 𝑧 is non-trivial. But then we get the commuting diagram
𝑃 (0) 𝑃 (0) 𝑃 (𝜆) 𝑃 (𝜆) 𝑓 𝑧=0 𝑓 𝑧
using the properties of central maps. But one of the compositions is not zero by the above reasoning and the other is zero, as we have assumed that 𝑧 acts trivially on 𝑃 (0). Thus we get a contradiction to the above assumption and the result follows.
Now we will prove the conditions of the Lemma are satisfied in our setting. It is well known from BGS [8], that the categories of perverse sheaves 𝑃𝐼0𝐼, 𝑃𝐼𝑊,𝐼
are indeed highest weight categories, where the standard objects are given by 𝑗𝑤! :=
(𝑗𝑤)!( ¯Q𝑙′[𝑙(𝑤)]), where 𝑗𝑤 : ℱ 𝑙𝑤 ˓→ ℱ 𝑙 is the inclusion of the 𝐼-orbit, respectively the
𝛿𝑤 = 𝐴𝑣𝐼𝑊(𝑗𝑤!) for 𝑤 ∈𝑓𝑊 . Similarly the categories 𝑃̃︁ 𝐼0𝐼0 and 𝑃𝐼𝑊 𝐼0 are categories with certain standard objects ˜𝑗𝑤!, respectively ˜∆𝑤 satisfying similar conditions except
they are no longer an artinian category.
We now prove the condition about the socle, which is satisfied on the nose in the first case and some similar condition is satisfied in the second case. To state this we note that the orbit corresponding to 𝑖𝑑 ∈ ̃︁𝑊 is a single point. We denote by 𝛿 the simple object supported at this point.
The following result and proof appear in Beilinson, Bezrukavnikov and Mirkovic [11] in a similar context, but we include it again for completeness
Lemma 9. For 𝑗𝑤!∈ 𝑃 𝑒𝑟𝑣𝐼0(ℱ 𝑙), the socle is simple and given by 𝛿.
Proof. We proceed by induction on length of 𝑤. For length 0 we get 𝑖𝑑 and the result is obvious.
the short exact sequence
0 → 𝛿 → 𝑗𝑠! → 𝐿𝑠 → 0
where 𝐿𝑠∼= Cℱ 𝑙𝑠[1] is the simple object. Thus the result also follows for length one.
For higher length 𝑤 let 𝑤 = 𝑤′𝑠 for 𝑠 a simple reflection of the simple root 𝛼 and 𝑙(𝑤′) = 𝑙(𝑤) − 1. We will denote by 𝑖𝑤 : 𝜋−1𝛼 (𝜋𝛼(ℱ 𝑙𝑤)) ˓→ ℱ 𝑙, where 𝜋𝛼 : ℱ 𝑙 → ℱ 𝑙𝛼
is the projection to the partial flag variety ℱ 𝑙𝛼 corresponding to minimal projective given by the simple root 𝛼.
Then note in fact, that 𝜋𝛼−1(𝜋𝛼(ℱ 𝑙𝑤)) is a P1 bundle over an affine space, thus is a
trivial bundle, so we can use a similar exact sequence like above to get
0 → 𝑗𝑤′!→ 𝑗𝑤! → 𝑖𝑤! → 0
where we use the notation 𝑖𝑤! just like above.
Now by induction hypothesis, we can assume the socle of 𝑗𝑤′! is given by 𝛿. To prove
the result now assume the 𝐿 is a simple in the socle of 𝑗𝑤!, if it lies in the image of 𝑗𝑤′!,
then the result follows by induction, so we assume that 𝐿 ˓→ 𝑖𝑤!. Note that as this is
constant along the smooth fibers of the fiber bundle 𝜋𝛼, we must have 𝐿 ∼= 𝜋*𝛼(𝐿′)[1]
for some simple perverse sheaf on ℱ 𝑙𝛼. But consider thus
𝐻𝑜𝑚(𝜋*𝛼(𝐿′)[1], 𝑗𝑤!) ∼= 𝐻𝑜𝑚(𝐿′[1], (𝜋𝛼)*(𝑗𝑤!)) ∼= 𝐻𝑜𝑚(𝐿′[1], 𝑗𝑤!𝛼[−1]).
Here 𝑗𝛼
𝑤! is the standard object in ℱ 𝑙𝛼 corresponding to the orbit of 𝑤. It follows
that the LHS Hom has to be 0 contradicting this being a subobject. Thus we get the result of the socle as required.
Remark 1. Note that the proof of the above Lemma, in fact shows that 𝛿 ˓→ 𝑗𝑤! has
cokernel satisfying that no 𝛿 appears as a simple subquotient.
Similarly in the category 𝑃 𝑒𝑟𝑣𝐼0(̃︁ℱ 𝑙) every object ˜𝑗𝑤! has the free cover ˜𝛿 as a
sub-object and every subsub-object intersects this sub-object. The proof follows by considering that ˜𝑗𝑤! are free monodromic objects, so we can reduce the argument to the above by
We will now use results proved by Arkhipov, Bezrukavnikov [2] and again in a more general context by Bezrukavnikov Yun [16]. We will then generalize the condition about the socle.
We first recall the functor 𝐴𝑣𝐼𝑊 : 𝑃𝐼0𝐼 → 𝑃𝐼𝑊 𝐼 which is given by convolution with
∆0. Similarly we have the functor 𝐴𝑣𝐼𝑊 : 𝑃𝐼0𝐼0 → 𝑃𝐼𝑊 𝐼0, which by abuse of notation
we denote the same.
We further have for both a right and left adjoint functors 𝐴𝑣𝐼0,!/*: 𝑃𝐼𝑊,𝐼0 → 𝑃𝐼0𝐼0 and
𝐴𝑣𝐼0,!/* : 𝑃𝐼𝑊,𝐼 → 𝑃𝐼0𝐼. As stated in Bezrukavnikov, Yun [16] we have 𝐴𝑣𝐼0,!(𝐴𝑣𝐼𝑊(−)) ∼=
Ξ * (−). Here Ξ is the projective cover of 𝛿 as an object of 𝑃 𝑒𝑟𝑣𝑁∨(𝐺∨/𝑁∨) of the
finite flag variety as constructed in that paper, where 𝑁∨ is the unipotent radical of 𝐵∨.
We now state several properties of these functors. Lemma 10. ∙ We have
𝐴𝑣𝐼𝑊(𝐿𝑤) = 0 ⇔ 𝑤 /∈𝑓̃︁𝑊 .
∙ 𝐴𝑣𝐼𝑊 is a t-exact functor with respect to the perverse t-structure.
∙ If 𝑤 = 𝑥𝑤′ where 𝑥 ∈ 𝑊 , 𝑤′ ∈𝑓
̃︁ 𝑊 , then
𝐴𝑣𝐼𝑊(𝑗𝑤!) ∼= ∆𝑤′.
A similar result holds for ˜𝑗𝑤! in 𝑃𝐼0𝐼0.
Proof. ∙ Note first that the functor 𝐴𝑣𝐼𝑊 can by defined by convolution with ˜∆0
as seen in Bezrukavnikov, Yun [16]. If 𝑤 ∈𝑓̃︁𝑊 , then the convolution diagram is an isomorphism on the generic part of the support. It thus follows that we get a non-zero object.
Now if 𝑤 /∈ 𝑓
̃︁
𝑊 , then 𝑠𝛼𝑤 < 𝑤 for some finite simple root 𝛼. Thus we get
that convolving with 𝐿𝑤 factors through (𝜋𝛼)* for 𝜋𝛼 : ℱ 𝑙 → ℱ 𝑙𝛼. So the result
will follow, if we prove (𝜋𝛼)*( ˜∆0) = 0. Note that along the fibers of 𝜋𝛼, the
∙ 𝐴𝑣𝐼𝑊 is given by 𝐴𝑣𝐼𝑊,! = 𝐴𝑣𝐼𝑊,* and it is the pushforward along an affine
morphism. So it follows from this that it is both left and right t-exact with respect to the perverse t-structure.
∙ For 𝑤 = 𝑥𝑤′ as in the statement of the Lemma, we have that 𝑗
𝑥𝑤′ = ˜𝑗𝑥!* 𝑗𝑤′!.
So we can reduce the result to the statement that 𝐴𝑣𝐼𝑊(˜𝑗𝑥!) = ˜∆0 for 𝑥 ∈ 𝑊 .
Now note that by the previous result we have a map ˜𝛿 ˓→ ˜𝑗𝑥! and the cokernel
has simple subquotients 𝐿𝑦 for 𝑦 ∈ 𝑊 r {𝑖𝑑}. Note that 𝐴𝑣𝐼𝑊 is exact and
sends all these simples to 0. Thus this map induces an isomorphism 𝐴𝑣𝐼𝑊(˜𝛿) ∼=
𝐴𝑣𝐼𝑊(˜𝑗𝑥!). Now the result follows, as the result for ˜𝛿 is clear.
The result from ˜𝑗𝑤! follows by the same result as above.
Now we prove the equivalent result of the socle
Lemma 11. The socle of ∆𝑤 is given by ∆0
Proof. Assume for contradiction that we have some irreducible subobject of ∆𝑤 that
is not ∆0. Note that every irreducible is given by 𝐴𝑣𝐼𝑊(𝐿𝑥) for some 𝑥. Thus by
adjunction we have 𝐿𝑥˓→ Ξ * 𝑗𝑤!.
But note that 𝑃 has a filtration with associated graded given by 𝑗𝑢! for some 𝑢 ∈ 𝑊 .
Thus we get that Ξ * 𝑗𝑤! has a filtration with associated graded given by 𝑗𝑢𝑤!. Note
that these have socles all given by 𝛿, so there is no non-zero morphism from any other simple.
Further the ∆0 appears with multiplicity 1 on ∆𝑤. This follows by exactness and the
fact that the similar result holds for 𝑗𝑤!.
Remark 2. In the case 𝑃𝐼𝑊 𝐼0 just as above we have a subobject ˜∆0 of ˜∆𝑤 such that
each subobject intersects this one.
Now the result about the center follows from both these categories as required. Now we need to identify the projective cover 𝒫 of 𝛿 and more concretely under-stand 𝐸𝑛𝑑(𝒫). For this we quote some results of Bezrukavnikov, Yun [16].
