Hecke Algebras and Finite Groups of Lie Type
Here we explain the fundamental connections between the theory of Iwahori-Hecke algebras and representations of a finite group of Lie type G. We study the modular representation theory of G and show how our previous results on “cell data” and “canonical basic sets” leads to a natural parametrization of the modular irreducible representations of G which admit non-zero vectors fixed by a Borel subgroup. This generalises classical results due to Bourbaki, Iwahori, Tits, which are concerned with the characteristic 0 situation. We then discuss a number of examples and open problems. This includes a conjectural classification of all the irreducible representations of G in the “non-defining characteristic case”.
KeywordsIrreducible Representation Finite Group Composition Factor Discrete Valuation Ring Cuspidal Representation
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