GENERIC AND q-RATIONAL REPRESENTATION THEORY

Part I of this paper develops various general concepts in generic representation and cohomology theories. Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of nite and algebraic groups, and we formalize the notion of a \generic" property in representation theory. Part II makes new contributions to the non-describing representation theory of nite general linear groups. First, we present an explicit Morita equivalence connecting GLn(q) with the theory of q-Schur algebras, extending a unipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equivalence to study the cohomology groups H (GLn(q); L), when L is an irreducible module in non-describing characteristic. The generic theory of Part I then yields stability results for various groups H(GLn(q); L), reminscent of our general theory [CPSK] with van der Kallen of generic cohomology in the describing characteristic case. (In turn, the stable value of such a cohomology group can be expressed in terms of the cohomology of the a ne Lie algebra b gln(C ).) The arguments entail new applications of the theory of tilting modules for q-Schur algebras. In particular, we obtain new complexes involving tilting modules associated to endomorphism algebras obtained from general nite Coxeter groups.