THE q-SCHUR ALGEBRA

We study a class of endomomorphism algebras of certain q-permutation modules over the Hecke algebra of type B, whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed. The notion of a q-Schur algebra was introduced by Dipper and James [DJ2], who used these algebras to parametrize the irreducible representations of the finite general linear groups in non-describing characteristics. With hindsight, these algebras had already appeared earlier in an entirely different quantum group context [Ji] inspired by physics. In [Ji] Jimbo considered the endomorphism algebras of tensor spaces as Hecke algebra modules. In his context, a q-Schur algebra can be viewed as a quotient of the quantized enveloping algebra associated to gln. In [PW], these algebras were shown to be quasi-hereditary. The quasi-heredity property is an embodiment in classical algebra of the geometric derived category stratification exhibited by perverse sheaves [PS]. It means more applications can be deduced from a ring-theoretic point of view (see e.g., [DPS3]), and the possibility is raised of even deeper results in the future, as suggested by [CPS2]. Certainly, these algebras play a central role in the representation theories of the finite and quantum general linear groups. Naturally, one asks: Are there such algebras for types other than A? Our paper [DS] showed that there were similar quasi-hereditary quotients of quantized enveloping algebras for all types of root systems. However, no connection with Hecke algebras and finite groups of Lie type was found there. This paper aims at the same question and constructs possible algebras directly from Hecke algebras (hence from finite groups of Lie type). We restrict attention to the type B case. Imitating the definition of a q-Schur algebra, we introduce the notion of a q-Schur algebra. These algebras are the endomorphism algebras of certain modules over the Hecke algebra of type B — called “tensor” spaces — whose summands involve not only parabolic subgroups but also quasi-parabolic subgroups of the Weyl group of type B. A main result of our paper shows that qSchur algebras are quasi-hereditary. We speculate that similar constructions exist for other classical types (only type D remains, actually, since type C is equivalent Received by the editors March 3, 1997 and, in revised form, October 28, 1998. 2000 Mathematics Subject Classification. Primary 20C08, 20G05, 20C33. The authors would like to thank ARC for support under the Large Grant A69530243 as well as NSF, and the Universities of Virginia and New South Wales for their cooperation. The first author also thanks the Newton Institute at Cambridge for its hospitality. c ©2000 American Mathematical Society