Parabolic character sheaves, II

粘合抛物特征层

Gluing parabolic character sheaves

该研究引入并研究伴随型约化群对应Vinberg半群非退化轨迹上幂幺特征层范畴，证明其偏对象可由抛物特征层粘合，应用中证明He的猜想并否定Lusztig关于Steinberg特征层中间扩张的猜想。

Character sheaves on disconnected groups, IX

We associate a two-sided cell to any (parabolic) character sheaf. We study the interaction of the duality operator for character sheaves and the operation of "twisted induction".

Character sheaves on certain spherical varieties

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.

Character sheaves and generalizations

This paper contains an exposition of the theory of character sheaves for reductive groups and some attempts to extend it to other cases: unipotent groups, reductive groups modulo the unipotent radical of a parabolic.

双轨道层与单位根处的仿射赫克代数

Bi-orbital sheaves and affine Hecke algebras at roots of unity

本文在 $\mathbb{Z}/m$-分次李代数幂零锥上引入超尖层概念，证明双轨道层由超尖层经抛物诱导产生，并据此证明了 Grojnowski 关于单位根处仿射赫克代数单模参数化的定理。

Categorification of a parabolic Hecke module via sheaves on moment graphs

We investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting and use it to categorify a parabolic Hecke module. As an application we obtain a combinatorial description of indecomposable projective objects of (truncated) non-critical singular blocks of (a deformed version of) category $\mathcal{O}$, using indecomposable special modules over the structure algebra of the corresponding Bruhat graph.

半无限抛物IC层

Semi-infinite parabolic IC-sheaf

研究连通约化群G及抛物子群P相关问题，通过构造仿射格拉斯曼流形上抛物半无限层范畴中各轨道的半无限IC层的抛物版本，建立其性质并与其他对象联系，实现多方面关联。

Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology

For a fixed parabolic subalgebra p of gl(n,C) we prove that the centre of the principal block O(p) of the parabolic category O is naturally isomorphic to the cohomology ring of the corresponding Springer fibre. We give a diagrammatic description of O(p) for maximal parabolic p and give an explicit isomorphism to Braden's description of the category Perv_B(G(n,n)) of perverse sheaves on Grassmannians. As a consequence Khovanov's algebra H^n is realised as the endomorphism ring of some object from Perv_B(G(n,n)) which corresponds under localisation and the Riemann-Hilbert correspondence to a ful…

Toward AGT for parabolic sheaves

We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$, and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.

A class of perverse sheaves on a partial flag manifold

We introduce a class of perverse sheaves on a partial flag manifold of a connected reductive group G defined over a finite field which are equivariant under the action of the group of rational points of G. The definition of this class is similar to the definition of parabolic character sheaves.

Character sheaves on disconnected groups, VIII

In this paper we continue the study of character sheaves on a reductive group G. To each subset of the set of simple reflections in the Weyl group we associate an algebra of the same kind as an Iwahori-Hecke algebra with unequal parameters in terms of parabolic character sheaves. We also prove a Mackey type formula for character sheaves. We define a duality operation for character sheaves. We also prove a quasi-rationality property for character sheaves.

特征层的导出范畴 II：典范诱导/限制函子

Derived categories of character sheaves II: canonical induction/restriction functors

本文推广了特征层 dg 范畴的组合描述至一般复约化群，并明确了抛物诱导/限制函子。

On the Deligne-Lusztig involution for character sheaves

For a reductive group G, we study the Drinfeld-Gaitsgory functor of the category of conjugation-equivariant D-modules on G. We show that this functor is an equivalence of categories, and that it has a filtration with layers expressed via parabolic induction of parabolic restriction. We use this to provide a conceptual definition of the Deligne-Lusztig involution on the set of isomorphism classes of irreducible character D-modules, which was defined previously in [Lu1, section 15].

Nonsplit Hecke algebras and perverse sheaves

Let H be a Hecke algebra arising as an endomorphism algebra of the representation of a Chevalley group G over F_q induced by a unipotent cuspidal representation of a Levi quotient L of a parabolic subgroup. We assume that L is not a torus. In this paper we outline a geometric interpretation of the coefficients of the canonical basis of H in terms of perverse sheaves. We illustrate this in detail in the case where the Weyl group of G is of type B_4 and that of L is of type B_2.

Parabolic Induction and Geometry of Orbital Varieties for GL(n)

Ariki and Ginzburg, after the previous work of Zelevinsky on orbital varieties, proved that multiplicities in a total parabolically induced representations are given by the value at q=1 of Kazhdan-Lusztig Polynomials associated to the symmetric groups. In my thesis I explore the geometry of orbital varieties and I essentially obtain two important results: - first I prove a conjecture of Zelevinksy on a property of independence of total parabolically induced representations. - More crucially I give a strategy to compute multiplicities in general parabolically induced representations using the p…

Study of parity sheaves arising from graded Lie algebra

Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspid…

Toward a Mackey formula for compact restriction of character sheaves

We generalize a result by Cunningham-Salmasian to a Mackey-type formula for the compact restriction of a semisimple perverse sheaf produced by parabolic induction from a character sheaf, under certain conditions on the parahoric group used to define compact restriction. This provides new tools for matching character sheaves with admissible representations.

Uhlenbeck spaces for $A^2$ and affine Lie algebra $\hat{sl}_n$

We introduce an Uhlenbeck closure of the space of based maps from projective line to the Kashiwara flag scheme of an untwisted affine Lie algebra. For the algebra $\hat{sl}_n$ this space of based maps is isomorphic to the moduli space of locally free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. The Uhlenbeck closure admits a resolution of singularities: the moduli space of torsion free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. We compute the Intersection Cohomology sheaf of the Uhlenbeck space using this resolution of singularities. The moduli spaces of par…

Parahoric子群上的Generic特征层

Generic character sheaves on parahoric subgroups

本文研究约化群环路群中parahoric子群概形上的抛物诱导，证明其在特定条件下产生共轭等变反常层并构成范畴等价，同时解决了Lusztig的一个猜想。

Notes on character sheaves

In the first section we study a functor of Bezrukavnikov, Finkelberg and Ostrik defined on character sheaves; we compute it in a Grothendieck group taking weights into account. In the second section we enlarge the class of character sheaves to a larger class of simple perverse sheaves which behaves well under tensor product (unlike the character sheaves themselves).

The character sheaves on the group compactification

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in \cite{L3}. We also prove some properties of the character sheaves on the group compactification.

Character sheaves on disconnected groups, V

We continue the study of character sheaves on a not necessarily connected reductive group. We prove orthogonality formulas for certain characteristic functions.

On the cleanness of cuspidal character sheaves

We prove the cleanness of cuspidal character sheaves in the few cases where it was previously unknown.

Character sheaves on disconnected groups, I

We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups.

Character sheaves on disconnected groups, VI

We define the notion of character sheaf on a possibly disconnected reductive group. We show that the restriction functor carries a character sheaf to a direct sum of character sheaves.

Character sheaves on disconnected groups, X

We classify the unipotent character sheaves on a fixed connected component of a reductive algebraic group under a mild hypothesis on the characteristic of the ground field.

在分级例外李代数上的尖锐特征sheaves：稳定分级

Cuspidal character sheaves on graded exceptional Lie algebras: stable gradings

本文研究了所有（GIT）稳定分级例外李代数的尖锐特征sheaves的显式确定方法，揭示了其核心贡献和数学结构。

沿$\mathrm{Bun}_G$的Harder-Narasimhan分层黏合层

Gluing sheaves along Harder-Narasimhan strata of $\mathrm{Bun}_G$

本文借助[FS21]构造的光滑图表局部闭分层上同调，给出沿Fargues-Scholze的$\mathrm{Bun}_G$的Harder-Narasimhan分层黏合素p挠层的方法，并在$G=\operatorname{GL}_2$时明确计算了几何常数项函子在特殊点的像。

Character sheaves on disconnected groups, III

We continue the attempt to develop a theory of character sheaves on a not necessarily connected reductive algebraic group. In this paper we introduce and study the generalized Green functions.

对称对的字符层：旋量群

Character sheaves for symmetric pairs: spin groups

本文研究旋量群对称对的字符层分类，确定了其中的尖点字符层，并证明它们可由近邻循环构造及其推广得到。

Non-unipotent character sheaves as a categorical centre

Let G be a reductive connected group over the algebraic closure of a finite field. In this paper we give the classification of character sheaves on G in categorical terms (as a categorical centre). Previously such a classification was known for unipotent character sheaves and also in the case where the ground field is replaced by the complex numbers.

特征层综述

A survey of character sheaves

本文综述了特征层理论的早期发展及近期进展，涵盖其在环群上的应用。

Derived categories of character sheaves

We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group $G$, similar to the one in generalized Springer correspondence. As a corollary, we identify the category of character sheaves on $G$ as the category of quasi-coherent sheaves on an explicitly defined derived stack $\widehat{G}$.

Generic character sheaves on disconnected groups and character values

We relate a generic character sheaf on a disconnected reductive group with a character of a representation of the rational points of the group over a finite field extending a result known in the connected case.

约化李代数上的模特征层

Modular Character Sheaves on Reductive Lie Algebras

本文将 Mirković 的李代数特征层理论推广至模系数，重证其部分结果，并建立与模广义 Springer 对应的联系。

Character sheaves on disconnected groups, II

The main theme of this paper is establishing the "generalized Springer correspondence" in complete generality that is, for not necessarily connected reductive algebraic groups.

凝聚层导出范畴上的 Co-$t$-结构与倾斜模的上同调

Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules

本文在幂零锥及抛物 Springer 解消的凝聚层导出范畴上构造 co-$t$-结构，用以刻画外尔奇偶对象推进的不可分解性，并证明 $A$ 型、$p>h$ 时相对 Humphreys 猜想的概形论形式。

Character sheaves on unipotent groups in characteristic p>0

These are slides for a talk given by the authors at the conference "Current developments and directions in the Langlands program" held in honor of Robert Langlands at the Northwestern University in May of 2008. The slides can be used as a short introduction to the theory of characters and character sheaves for unipotent groups in positive characteristic, developed by the authors in a series of articles written between 2006 and 2011. We give an overview of the main results of this theory along with a bit of motivation.

约化群的幂单特征层与层化，II

Unipotent character sheaves and strata of a reductive group, II

本文将幂单特征层到层化的满射构造从连通约化群推广到约化群的连通分支，扩展了先前的研究结果。

Parahoric 特征层的比较定理

Characteristic Functions of Parahoric Character Sheaves

本文为 parahoric 子群上的特征层建立 Jordan 分解公式，证明其与深层 Deligne--Lusztig 特征的关系及正交性，并给出特征限制的层论表达式。

Character sheaves and characters of unipotent groups over finite fields

Let G_0 be a connected unipotent algebraic group over a finite field F_q, and let G be the unipotent group over an algebraic closure F of F_q obtained from G_0 by extension of scalars. If M is a Frobenius-invariant character sheaf on G, we show that M comes from an irreducible perverse sheaf M_0 on G_0, which is pure of weight 0. As M ranges over all Frobenius-invariant character sheaves on G, the functions defined by the corresponding perverse sheaves M_0 form a basis of the space of conjugation-invariant functions on the finite group G_0(F_q), which is orthonormal with respect to the standar…

Singular supports for character sheaves on a group compactification

Let $G$ be a semisimple adjoint group over $\bold C$ and $\bar{G}$ be the De Concini-Procesi completion of $G$. In this paper, we define a Lagrangian subvariety $Λ$ of the cotangent bundle of $\bar{G}$ such that the singular support of any character sheaf on $\bar{G}$ is contained in $Λ$.

Character sheaves on disconnected groups, IV

Let G be a possibly disconnected reductive group over a finite field with Frobenius map F. The main result of this paper is that the characteristic functions af "admissible complexes" A on G such that F^*A is isomorphic to A form a basis of the space of functions on G^F that are constant on G^{0F}-conjugacy classes.

幂单特征层与约化群的层

Unipotent character sheaves and strata of a reductive group

本文通过广义 Springer 对应构造了连通约化群上幂单特征层到层的满射，并利用坏特征数据给出了幂单特征层的新参数化。

正特征下幂幺群上的特征层：基础

Character sheaves on unipotent groups in positive characteristic: foundations

研究正特征下幂幺群上特征层理论，证明其主要定理，即容许对与特征层的\(L\)-包的对应关系，还通过附录探讨了相关范畴理论模式，如幺半范畴对偶、拓扑场论方法定义结构及证明导出范畴定义等价等。

Character sheaves on neutrally solvable groups

Let $G$ be an algebraic group over an algebraically closed field $\mathtt{k}$ of characteristic $p>0$. In this paper we develop the theory of character sheaves on groups $G$ such that their neutral connected components $G^\circ$ are solvable algebraic groups. For such algebraic groups $G$ (which we call neutrally solvable) we will define the set $\operatorname{CS}(G)$ of character sheaves on $G$ as certain special (isomorphism classes of) objects in the category $\mathscr{D}_G(G)$ of $G$-equivariant $\overline{\mathbb{Q}}_\ell$-complexes (where we fix a prime $\ell\neq p$) on $G$. We will desc…

On the unipotent support of character sheaves

Let $G$ be a connected reductive group over $F_q$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of {\em character sheaves}, a geometric version of the classical character theory of the finite group $G(F_q)$. We show that under a certain technical condition, the restriction of a character sheaf to its {\em unipotent support} (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\Z$-basis of the $\Z$-module of unipotently supported virtual char…

Constructible sheaves on nilpotent cones in rather good characteristic

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, inclu…

Truncated convolution of character sheaves

Let G be a reductive connected group over an algebraic closure of a finite field. I define a tensor structure on the category of perverse sheaves on G which are direct sums of unipotent character sheaves in a fixed two-sided cell, in accordance with a conjecture I have made in 2004. I also show that that the resulting monoidal category is equivalent to the centre of a monoidal category which I defined in 1997 (a categorical version of the J-ring attached to the same two-sided cell), thus verifying a conjecture of Bezrukavnikov, Finkelberg, Ostrik. A possible interpretation of unipotent charact…

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good chara…

Sheaves on the alcoves and modular representations II

We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.

Reductive groups, the loop Grassmannian, and the Springer resolution

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic $p$ bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application…

Exotic symmetric space over a finite field, III

Let X be an exotic symmetric space G/H \times V, where G= GL(V) and H = Sp(V) for a symplectic vector space V over an algebraically closed field of odd characteristic. In our previous papers, character sheaves on X were constructed based on the explicit data. In this paper, we give a conceptual definition of character sheaves on X based on the idea of Ginzburg in the case of symmetric spaces, which we call Ginzburg type character sheaves. We prove that Ginzburg type character sheaves on X actually coincide with those defined in our papers. This gives a classification of Ginzburg type character…

On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front se…