# Publications of the Research Institute for Mathematical Sciences

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**Volume 35, Issue 1, 1999, pp. 31–90**

**DOI: 10.2977/prims/1195144189**

Generic and *q*-Rational Representation Theory

^{[1]}, Brian Parshall

^{[2]}and Leonard Scott

^{[3]}(1) Department of Mathematics, University of Oklahoma, OK 73019-0315, NORMAN, UNITED STATES

(2) Department of Mathematics, University of Virginia, P.O. Box 400137, VA 22904-4137, CHARLOTTESVILLE, UNITED STATES

(3) Department of Mathematics, University of Virginia, P.O. Box 400137, VA 22904-4137, CHARLOTTESVILLE, UNITED STATES

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 *GL _{n}*(

*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*(

*GL*(

_{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*

^{1}(

*GL*(

_{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

**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.

*No keywords available for this article.*

Cline E, Parshall B, Scott L. Generic and *q*-Rational Representation Theory. *Publ. Res. Inst. Math. Sci.* 35 (1999), 31-90. doi: 10.2977/prims/1195144189