STRATIFYING ENDOMORPHISM ALGEBRAS ASSOCIATED TO HECKE ALGEBRAS

Let G be a finite group of Lie type and let k be a field of characteri stic distinct from the defining characteristic of G. In studying the n on-describing representation theory of G, the endomorphism algebra S(G , k) = End(kG)(circle plus(J) ind(PJ)(G)k) plays an increasingly impor tant role. In type A, by work of Dipper and James, S(G, k) identifies with a q-Schur algebra and so serves as a link between the representat ion theories of the finite general linear groups and certain quantum g roups. This paper presents the first systematic study of the structure and homological algebra of these algebras for G of arbitrary type. Be cause S(G, k) has a reinterpretation as a Hecke endomorphism algebra, it may be analyzed using the theory of Hecke algebras. Its structure t urns out to involve new applications of Kazhdan-Lusztig cell theory. I n the course of this work, we prove two stratification conjectures abo ut Coxeter group representations made by E. Cline, B. Parshall, and L. Scott (Mem. Amer. Math. Sec. 591, 1996) and we formulate a new conjec ture about the structure of S(G, k). We verify this conjecture here in all rank 2 examples. (C) 1998 Academic Press.