# Generic and <em>q</em>-Rational Representation Theory

### Edward Cline

University of Oklahoma, Norman, USA### Brian Parshall

University of Virginia, Charlottesville, USA### Leonard Scott

University of Virginia, Charlottesville, USA

## 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 *GLn*(*q*) with the theory of *g*-Schur algebras, extending a unipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equivalence to study the cohomology groups em>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_1(*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 affine Lie algebra **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.

## Cite this article

Edward Cline, Brian Parshall, Leonard Scott, Generic and <em>q</em>-Rational Representation Theory. Publ. Res. Inst. Math. Sci. 35 (1999), no. 1, pp. 31–90

DOI 10.2977/PRIMS/1195144189