STRATIFYING q-SCHUR ALGEBRAS OF TYPE D

Two families of q-Schur algebras associated to Hecke algebras of type D are introduced, and related to a family used by Geck, Gruber and Hiss [10], [11]. We prove that the algebras in one family, called the q-Schur algebras, are integrally free, stable under base change, and are standardly stratified if the base field has odd characteristic. In the so-called linear prime case of [10],[11], all three families give rise to Morita equivalent algebras. A final section discusses a different example, and speculates on the direction of a general theory. Following up our recent work [9] for type B (and C), we introduce here some endomorphism algebras of type D. These are of possible use in determining irreducible representations of finite groups of Lie type D, especially in non-defining characteristics, as in the work of GeckHiss [10] and Gruber-Hiss [11] following the spirit of Dipper-James’ work in type A. (See Remark 2.11 below for further discussion.) We organize the paper as follows: Section 1 sets up preparation on Weyl groups of classical types and distinguished coset representatives, especially those with the trivial intersection property. The various q-Schur algebras are introduced in Section 2 and connections between them are also discussed. The linear prime Morita equivalence theorems are proved in Section 3. In Sections 4, 5 and 6, we establish further results on q-permutation and twisted q-permutation modules of type B. These results are supplementary to those given in [9]. In particular, we prove, in the type B setting, the homological property required (see [7]) in stratifying an endomorphism algebra for type D. The main results are given in Section 7, where the quasi-heredity in the odd degree case is proved for one of our algebras, called the q-Schur algebra. A standard stratification in the even degree case is constructed for the same algebra when 2 is invertible in the base ring. Section 8 gives an effective approach to the bad prime case p = 2, and concludes with speculations regarding a general theory. Date: 6 December, 1999. 1991 Mathematics Subject Classification. 20G05, 16S80. The authors would like to thank ARC for support as well as NSF, and the Universities of Virginia and New South Wales for their cooperation.