Titlebook: Representations of SL2(Fq); Cédric Bonnafé Textbook 2011 Springer-Verlag London Limited 2011 Deligne-Lusztig theory.Morita equivalences.SL

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Unequal Characteristic: Equivalences of Categories abelian defect group), the equivalences of categories predicted by Broué’s conjecture are always Morita equivalences (see Sections?8.1 and?8.2). While it is possible to obtain this result using Brauer trees and Brauer’s theorem?B.4.2, we give instead a concrete construction of these equivalences us

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Equal Characteristic to the construction of the simple .-modules. This classical construction generalises to the case of finite reductive groups. It turns out that the simple .-modules are the restrictions of simple “rational representations” of the algebraic group .. Having obtained this description the determination

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Structure of ,The purpose of this chapter is to study the structure of the group .: noteworthy subgroups (tori, Borel subgroups, the Bruhat decomposition), distinguished subgroups, conjugacy classes, Sylow subgroups and their normalisers.

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The Geometry of the Drinfeld CurveThe purpose of this chapter is to assemble the geometric properties of . and of the action of .×(..?〈.〉.) which allows us to calculate its .-adic cohomology (as a module for the monoid .×(..?〈.〉.)). A large part of this chapter is dedicated to the construction of quotients of . by the actions of the finite groups ., . and ...

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Harish-Chandra InductionIn this chapter we study Harish-Chandra induction, which associates to a .-module the .-module obtained by first extending the .-module to a .-module (letting . act trivially) and then inducing to .. This construction allows us to obtain roughly half of the irreducible characters of ..

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Special CasesIn this chapter we will make explicit certain exotic properties of the groups . when .=3, 5 or 7. These include exceptional isomorphisms, inclusions as subgroups of ., and realisations as subgroups of reflection groups. For a recollection of definitions, results about reflection groups, see the Appendix C.

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