Deligne–Lusztig theory
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| Deligne–Lusztig theory | |
|---|---|
| Name | Deligne–Lusztig theory |
| Discipline | Mathematics |
| Subdiscipline | Algebraic geometry; Representation theory; Number theory |
| Developed | 1970s |
| Key figures | Pierre Deligne; George Lusztig; Jean-Pierre Serre; Alexander Grothendieck; John Tate |
Deligne–Lusztig theory is a framework originating in the work of Pierre Deligne and George Lusztig that constructs representations of finite groups of Lie type using algebraic varieties over finite fields and cohomological techniques. The theory links ideas from algebraic geometry, representation theory, and number theory to produce virtual representations, characterize irreducible characters, and relate geometric objects to the representation theory of groups such as GL(n), SL(n), Sp(2n), SO(n), and exceptional groups like G2, F4, E6, E7, E8. It employs tools developed by figures such as Alexander Grothendieck, Jean-Pierre Serre, John Tate, and later contributions from Robert Langlands and Gérard Laumon.
Introduction
Deligne–Lusztig theory arose from the problem of classifying irreducible representations of finite reductive groups, influenced by the earlier work of Claude Chevalley, Ernst Witt, Armand Borel, and Jacques Tits. The approach uses varieties defined over finite fields in the spirit of the Grothendieck school and applies étale cohomology methods pioneered by Grothendieck and Alexander Grothendieck's collaborators. The foundational paper by Deligne and Lusztig connects this geometric viewpoint to the character theory developed by Ferdinand Frobenius, Issai Schur, and later advanced by G. I. Lehrer and Roger Carter.
Construction of Deligne–Lusztig Varieties
Deligne–Lusztig varieties are constructed from data attached to a connected reductive algebraic group G defined over a finite field, a maximal torus T, and a Frobenius endomorphism F as in the work of Armand Borel and Jacques Tits. The varieties live in flag varieties associated to G à la Claude Chevalley and use parabolic subgroups and Borel subgroups studied by Jean-Pierre Serre and Robert Steinberg. The construction parallels ideas from the theory of algebraic groups by T. A. Springer and the Bruhat decomposition associated to N. Bourbaki and Michel Demazure. Frobenius-fixed points and the action of the finite group G(F_q) relate to counting points techniques similar to those in the Weil conjectures studied by André Weil and resolved by Pierre Deligne.
Representations of Finite Reductive Groups
Cohomology of Deligne–Lusztig varieties yields virtual representations of finite groups of Lie type such as GL(n, F_q), SL(n, F_q), Sp(2n, F_q), and groups introduced by Robert Steinberg and Claude Chevalley. George Lusztig used these constructions to parametrize most irreducible representations in series indexed by conjugacy classes of tori, building on the classification efforts of Roger Carter and conjectures by Robert Langlands about correspondences. The interaction of these series with unipotent classes studied by David Kazhdan and George Lusztig himself produces families reflecting phenomena earlier found in the work of Emil Artin and Frobenius.
Character Formulas and Applications
Deligne–Lusztig characters are computed using Lefschetz trace formulas in the style of Grothendieck and employ character sheaves introduced by George Lusztig with antecedents in the work of Robert Kottwitz and Mark Reeder. Applications include explicit character tables for groups studied by F. D. Murnaghan and Gerald Murphy, connections to the local Langlands program proposed by Robert Langlands, and implications for the theory of automorphic forms as explored by James Arthur and Gérard Laumon. The resulting formulas bridge to computational projects handled by researchers in the tradition of John Thompson and Bertram Huppert.
Cohomological Methods and Étale Cohomology
Étale cohomology, as developed by Alexander Grothendieck and formalized by Jean-Pierre Serre and Pierre Deligne, underpins the computation of intersection cohomology and perverse sheaves on Deligne–Lusztig varieties. The formalism of derived categories and perverse sheaves by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne interacts with the Springer correspondence developed by T. A. Springer and impacts the study of character sheaves and intersection complexes used by George Lusztig. These cohomological tools link to the work of Michael Artin and Grothendieck on the Lefschetz trace formula and to numerical invariants studied by André Weil.
Examples and Explicit Computations
Explicit Deligne–Lusztig varieties for groups like GL(2), SL(2), Sp(4), and G2 have been computed by authors building on computations of character tables by Frank Alperin, David Benson, and Roger Carter. Case studies include the Ree groups and Suzuki groups examined by Rimhak Ree and Michio Suzuki, and exceptional group computations in the tradition of G. E. Wall and Bertram Huppert. These explicit examples illuminate connections with modular representation theory explored by J. L. Alperin and congruences studied by Jean-Pierre Serre.
Extensions and Generalizations
Extensions of Deligne–Lusztig theory include parabolic Deligne–Lusztig constructions influenced by C. W. Curtis and I. Reiner, generalizations to disconnected groups studied by G. Malle and G. Lusztig, and relationships with affine Hecke algebras and Kac–Moody groups as developed by Victor Kac and Iwahori-Matsumoto frameworks. Further generalizations connect to the geometric Langlands program advocated by Robert Langlands, Edward Frenkel, and Gérard Laumon, and interact with categorical representation theory advanced by Maxim Kontsevich and Henri Cartan-inspired schools. Ongoing research by communities around institutions such as the Institute for Advanced Study, École Normale Supérieure, and Clay Mathematics Institute continues to expand the reach of these methods.