PRESENTING GENERALIZED q-SCHUR ALGEBRAS

We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztig’s modified form : U of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from : U, that this gives them a cellular structure, and thus they are quasihereditary over a field. Introduction In [Do1] Donkin defined the notion of a generalized Schur algebra for an algebraic group, depending on the group and a finite saturated subset π of dominant weights. He also showed how to construct the generalized Schur algebra from the enveloping algebra of the complex Lie algebra of the same type as the given algebraic group, by an appropriate modification of the construction of Chevalley groups. The purpose of this paper is to give a presentation by generators and relations for generalized Schur algebras and their quantizations. The presentation has the same form as [DG, Theorems 1.4, 2.4], for Schur algebras and q-Schur algebras in type A. We approach this problem the other way around. First we define (in §1) an algebra S(π) (over Q(v), v an indeterminate) by generators and relations. It depends only on a Cartan matrix (of finite type) and a given saturated set π of dominant weights. We prove that this algebra is a finite-dimensional semisimple quotient of the quantized enveloping algebra U determined by the Cartan matrix, and that it is a q-analogue of a generalized Schur algebra in Donkin’s sense. We show that S(π) is isomorphic with the algebra . U/ . U[P ] constructed by Lusztig in [Lu], where P is the complement of π in the set of dominant weights and where . U is his “modified form” of U, using his “refined Peter-Weyl theorem,” which for convenience we recall in §2. It follows that S(π) inherits a canonical basis and a cell datum (in the sense of Graham and Lehrer [GL]) from . U. This provides S(π) with an extremely rigid structure, which essentially determines its representation theory in all possible specializations. In particular, RS(π) is quasihereditary over any field R. These results are contained within §§3–5. In §6 we consider the classical case. First we define an algebra S(π) (over Q) by generators and relations. The defining presentation of S(π) is obtained from the Received by the editors August 31, 2002. 2000 Mathematics Subject Classification. Primary 17B37, 16W35, 81R50.