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Generic and q-Rational Representation TheoryEdward Cline, Brian Parshall and Leonard Scott (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 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 H1(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 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 finite Coxeter groups.
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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