Character sheaves on disconnected groups, VIII

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.

Character sheaves on disconnected groups, VII

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.

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.

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.

The Bernstein Center of a p-adic Unipotent Group

Francois Rodier proved that it is possible to view smooth representations of certain totally disconnected abelian groups (the underlying additive group of a finite-dimensional p-adic vector space, for example) as sheaves on the Pontryagin dual group. For nonabelian totally disconnected groups, the appropriate dual space necessarily includes representations which are not one-dimensional, and does not carry a group structure. The general definition of the topology on the dual space is technically unwieldy, so we provide three different characterizations of this topology for a large class of tota…

On Lusztig's conjecture for connected and disconnected exceptional groups

Lusztig conjectured that the almost characters of a finite reductive group are up to a scalar the same as the characteristic functions of the rational character sheaves defined on the corresponding algebraic group. We propose in this paper to verify this conjecture for the uniform almost characters of the Suzuki and the Ree groups and to show that a conjecture of the same type holds for the disconnected case of type B_2. We then state a conjecture on the possible values of the uniform almost characters on unipotent elements in the disconnected case of type F_4.

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…

Graded Hecke algebras for disconnected reductive groups

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G. We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come d…

Modules over etale groupoid algebras as sheaves

The author has previously associated to each commutative ring with unit $\Bbbk$ and étale groupoid $\mathscr G$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk$-algebra $\Bbbk\mathscr G$. The algebra $\Bbbk\mathscr G$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary $\Bbbk\mathscr G$-modules is equivalent to the category of sheaves of $\Bbbk$-modules over $\mathscr G$. As a consequence we obtain a new proof …

分次 Hecke 代数与幂零锥上等变可构造层

Graded Hecke algebras and equivariant constructible sheaves on the nilpotent cone

本文证明几何分次 Hecke 代数可实现为幂零锥上等变半单纯层复形的自同态代数，并将相应等变可构造层导出范畴等价描述为分次 Hecke 代数直和上的 DG 模导出范畴，从而给出其范畴化。

Crossed S-matrices and Character Sheaves on Unipotent Groups

Let $\mathtt{k}$ be an algebraic closure of a finite field $\mathbb{F}_{q}$ of characteristic $p$. Let $G$ be a connected unipotent group over $\mathtt{k}$ equipped with an $\mathbb{F}_q$-structure given by a Frobenius map $F:G\to G$. We will denote the corresponding algebraic group defined over $\mathbb{F}_q$ by $G_0$. Character sheaves on $G$ are certain objects in the triangulated braided monoidal category $\mathscr{D}_G(G)$ of bounded conjugation equivariant $\bar{\mathbb{Q}}_l$-complexes (where $l\neq p$ is a prime number) on $G$. Boyarchenko has proved that the "trace of Frobenius" funct…

Shintani descent for algebraic groups and almost characters of unipotent groups

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_q$. For this, it is essential to treat all the pure inner $\mathbb{F}_q$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the "trace of Frobenius" functi…

Springer 对应的推广与尖点 Langlands 参数

Generalizations of the Springer correspondence and cuspidal Langlands parameters

本文为约化 p-进群引入增强 Langlands 参数的尖点性概念及 Bernstein 分量，揭示其空间为扭曲扩展商并集，并将 Lusztig 广义 Springer 对应推广到不连通复约化群，从而将局部 Langlands 对应问题归结为 Levi 子群的超尖点表示情形。