G. Lusztig
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.
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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.
We study a class of perverse sheaves on the variety of pairs (P,gU_P) where P runs through a conjugacy class of parabolics in a connected reductive group G and gU_P runs through G/U_P. This is a generalization of the theory of 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).
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.
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.
We continue the study of character sheaves on a not necessarily connected reductive group. We prove orthogonality formulas for certain characteristic functions.
We prove the cleanness of cuspidal character sheaves in the few cases where it was previously unknown.
We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups.
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.
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.
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".
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。
Parabolic character sheaves and Hecke algebras
本文证明参数不等的Iwahori-Hecke代数可在抛物特征层框架中实现。
The theory of character sheaves on a reductive group is extended to a class of varieties which includes the strata of the De Concini-Procesi completion of an adjoint group.
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.
We define and study convolution of parabolic character sheaves. As an application we attach to any parabolic character sheaf the orbit of a tame local system on the maximal torus under a subgroup of the Weyl group.
We define character sheaves on an ind-variety of the form G((t))/U_P where G((t)) is a loop group and U_P is the prounipotent radical of a parahoric subgroup P of G((t)).
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。
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.
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。
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 $\widehat{G}$.
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.
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。
The main theme of this paper is establishing the "generalized Springer correspondence" in complete generality that is, for not necessarily connected reductive algebraic groups.
In this paper we investigate the representations of reductive groups over a finite field, introduced in 1987 by D.Kazhdan and G.Laumon. We show that generically these representations are irreducible and that their character is equal to the function obtained by traces of Frobenius morphism on the stalks of the corresponding Lusztig's character sheaf. This, together with the corresponding result by G.Lusztig, implies that generic Kazhdan-Laumon representations are isomorphic to the corresponding Deligne-Lusztig representations. In the second half of the paper we give a direct geometric construct…
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。
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.
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。
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。
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…
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 $Λ$.
We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t-exactness property of the Harish-Chandra transform and provide an evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.
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.
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。
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。
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…
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…
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…
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…
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…
Cuspidal character sheaves on graded Lie algebras II
本文研究了有级数李代数上尖点特征sheaves的分类及其支持，为明确分类这些sheaves提供了重要进展。
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.
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…
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…
In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.
Character sheaves for classical symmetric pairs
本研究为经典对称对构建Springer理论，明确刻画该场景下的特征层并确定尖点特征层，推进了相关表示论研究。
Invariant systems and character sheaves for graded Lie algebras
该研究通过引入微分方程不变系统构造分次李代数上的特征层，并与既有结果共同给出Vinberg I型经典情形的全部尖点特征层。
Character sheaves in characteristic $p$ have nilpotent singular support
本研究引入驯顺反常层范畴并研究其性质，证明任意特征下特征层具有幂零奇异支集，将相关学者的工作部分推广到正特征领域。
We study the extension groups between (modular) Wakimoto sheaves in type $\tilde{A_1}$. Firstly we determine them completely over characteristic zero fields. Secondly we describe a dg model which allows us to compute these groups in the antispherical category for arbitrary coefficients.
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.
Let $G$ be a unipotent group over a field of characteristic $p > 0$. The theory of character sheaves on $G$ was initiated by V. Drinfeld and developed jointly with D. Boyarchenko. They also introduced the notion of $\mathbb{L}$-packets of character sheaves. Each $\mathbb{L}$-packet can be described in terms of a modular category. Now suppose that the nilpotence class of $G$ is less than $p$. Then the $\mathbb{L}$-packets are in bijection with the set $\mathfrak{g}^*/G$ of coadjoint orbits, where $\mathfrak{g}$ is the Lie ring scheme obtained from $G$ using the Lazard correspondence and $\mathf…
Let $G$ be a connected reductive group over $\kk$, an algebraic closure of a finite field. For an integer $r\ge 1$ let $G_r=G(\kk[\e]/(\e^r))$ viewed as an algebraic group of dimension $r\dim G$ over $\kk$. We show that the character of the generic principal series representation of $G_r(F_q)$ can be realized by a simple perverse sheaf on $G_r$ provided that $r=2$ or $r=4$ and we give a strategy to prove the same statement for any even $r$. (The case where $r=1$ is already known.)
This paper is an introduction, in a simplified setting, to Lusztig's theory of character sheaves. It develops a notion of character sheaves on reductive Lie algebras which is more general then such notion of Lusztig, and closer to Lusztig's theory of character sheaves on groups. The development is self contained and independent of the characteristic $p$ of the ground field. The results for Lie algebras are then used to give simple and uniform proofs for some of Lusztig's results on groups.
This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when…
Character Sheaves for Graded Lie Algebras: Stable Gradings
本研究针对稳定分次李代数，通过分析[GVX2]中邻近周期层的傅里叶变换，构造出推测为尖点特征层的满支撑特征层，涉及单位根处复反射群相关Hecke代数的不可约表示。
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 for classical graded Lie algebras
本文针对源自特殊线性群内自同构的分次李代数与Vinberg II型典型分次李代数，开展其特征层的研究工作。
Character sheaves for symmetric pairs: special linear groups
本研究利用[VX1]的通用策略与中心特征分析，明确刻画了特殊线性群内对合对应的对称对的特征层，并确定了其中的尖点特征层。
We show that some types for supercuspidal representations of tamely ramified $p$-adic groups that appear in Jiu-Kang Yu's work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commut…
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.
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