JournalsprimsVol. 35, No. 1pp. 31–90

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
Generic and <em>q</em>-Rational Representation Theory cover
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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 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.

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