Generic and $q$-Rational Representation Theory

Abstract

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 finite 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 finite general linear groups. First, we present an explicipt Morita equivalence connecting $G{L}_n(q)$ 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 $H^{\ast }(G{L}_n(q),L)$, when $L$ is an irreducible module in non-describing characteristic. The generic theory of Part I then yields stability results for various groups $H^t(G{L}_n(q),L)$, 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 affine Lie algebra ${\widehat{\mathfrak{gl}}}_n(C)$.) 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 finite Coxeter groups.