Stratifying Endomorphism Algebras Associated to Hecke Algebras

Abstract

LetGbe a finite group of Lie type and letkbe a field of characteristicdistinctfrom the defining characteristic ofG. In studying thenon-describingrepresentation theory ofG, the endomorphism algebraS(G,k)=EndkG(⊕JindGPJk) plays an increasingly important role. In typeA, by work of Dipper and James,S(G,k) identifies with aq-Schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups. This paper presents the first systematic study of the structure and homological algebra of these algebras forGof arbitrary type. BecauseS(G,k) has a reinterpretation as a Hecke endomorphism algebra, it may be analyzed using the theory of Hecke algebras. Its structure turns out to involve new applications of Kazhdan–Lusztig cell theory. In the course of this work, we prove two stratification conjectures about Coxeter group representations made by E. Cline, B. Parshall, and L. Scott (Mem. Amer. Math. Soc.591, 1996) and we formulate a new conjecture about the structure ofS(G,k). We verify this conjecture here in all rank 2 examples.