character sheaves
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| character sheaves | |
|---|---|
| Name | Character sheaves |
| Field | Algebraic geometry; Representation theory |
| Introduced | 1980s |
| Key figures | George Lusztig, Pierre Deligne, David Kazhdan, Michael Artin, Alexandre Grothendieck |
character sheaves. Character sheaves are perverse sheaves on algebraic groups introduced to study characters of finite groups of Lie type. They connect the work of George Lusztig with concepts from Pierre Deligne's theories, bridging objects studied by Alexander Grothendieck's school, techniques of David Kazhdan, and advances influenced by Michael Artin and the Institute for Advanced Study. Their development transformed approaches in representation theory of Chevalley groups, Reed–Solomon code-adjacent counting problems, and interactions with the Langlands program, the Weil conjectures, and Kazhdan–Lusztig theory.
Introduction
Character sheaves are complexes of ℓ-adic sheaves (or D-modules) on a reductive algebraic group that categorize traces of Frobenius and produce class functions on finite groups like GL_n(F_q), SL_n(F_q), Sp_2n(F_q), and E_8(F_q). The theory synthesizes ideas from the work of Grothendieck on étale cohomology, Deligne on weights, and Lusztig on character formulas, while interacting with the frameworks of Harish-Chandra's admissible representations, Bernstein's center, and the categorical methods of Beilinson and Drinfeld.
Historical development
The initiative arose in Lusztig's program of the 1980s, building on earlier milestones: Grothendieck's formulation of étale cohomology, Deligne's proof of the Weil conjectures, and the formulation of perverse sheaves by Goresky and MacPherson. Influences also trace to work by Kazhdan and Lusztig on representations of affine Hecke algebras, and to categorical perspectives advanced by Beilinson, Bernstein, Drinfeld, and Verdier. Subsequent refinements connected with the Langlands dual group, results of Springer on Weyl group actions, and computations involving exceptional groups such as G_2, F_4, E_6, E_7, and E_8.
Definition and construction
One constructs character sheaves using methods from étale cohomology and perverse sheaves on a connected reductive group G over a finite field F_q, invoking the machinery of Deligne's weights and Verdier duality developed by Verdier. The basic ingredients include the Springer resolution associated to Tits's theory, nearby and vanishing cycles from Kashiwara and Schapira, and convolution functors inspired by Beilinson and Bernstein. The classification uses centralizers of semisimple elements following Steinberg's work, local systems on conjugacy classes tied to Lusztig's families, and endoscopic transfer related to Langlands duality. Construction variants employ D-modules in characteristic zero as in approaches by Hotta, Kashiwara, and Beilinson–Bernstein localization.
Relationship to representation theory
Character sheaves yield functions on G(F_q) via trace of Frobenius, producing almost characters that match irreducible character values studied by Deligne–Lusztig induction and the theory of Harish-Chandra characters. The framework interfaces with harmonic analysis on p-adic groups as in work by Bernstein, Casselman, and Rodier, and with the modular representation theory considered by Jantzen and Alperin. Connections to Hecke algebras emerge through Iwahori–Matsumoto constructions and affine Hecke algebra results by Lusztig and Kazhdan. The categorical perspective resonates with modern formulations of the geometric Langlands correspondence developed by Beilinson–Drinfeld and Gaitsgory.
Examples and computations
Explicit computations appear for classical series such as GL_n, SL_n, SO_n, and Sp_2n where character sheaves correspond to simple perverse sheaves supported on regular conjugacy classes or induced from Levi subgroups following Mackey-type formulas. Lusztig's tables for groups like E_8(F_q), F_4(F_q), and G_2(F_q) enumerate families and almost characters; these rely on intricate calculations using Weyl group combinatorics and Springer correspondence originally developed by Springer and Shoji. Tools from Deligne–Lusztig varieties, Kazhdan–Lusztig polynomials, and computations by Shoji and Spaltenstein produce explicit character values for unipotent classes.
Geometric and categorical properties
Character sheaves form semisimple categories closed under convolution and Verdier duality, reflecting properties analogous to the category O studied by Bernstein–Gelfand–Gelfand and to modular tensor categories appearing in Reshetikhin–Turaev constructions. Their supports relate to nilpotent orbits classified by Dynkin diagrams and studied by Kostant and Bala–Carter. Monodromy and local system data invoke concepts from Deligne's local monodromy theory and from the work of Simpson on nonabelian Hodge theory. Functorialities include restriction to Levi subgroups and induction via parabolic correspondences related to Mackey theory and Harish-Chandra induction.
Applications and extensions
Applications range across number theory, geometric representation theory, and mathematical physics: comparisons with automorphic trace formulas in the Langlands program, contributions to counting points on character varieties studied by Hausel and Letellier, and roles in mirror symmetry contexts influenced by Kontsevich. Extensions include categorifications linked to Khovanov homology, relations with categorical Langlands by Gaitsgory and Arinkin, and adaptations to quantum groups as in work by Drinfeld and Jimbo. Ongoing research explores equivariant enhancements, p-adic analogues studied by Scholze, and interactions with derived algebraic geometry developed by Lurie and Toen.