here l(w) is the length of w In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functions on the flag manifold of the corresponding finite Chevalley group G(Fq) (see [loc cit, Ex 24]) Therefore, the problem of decomposing this space of functions into irreducible representations of G(Fq) is equivalent to the problem of decomposing the regular representation o f ~ | (12 It is known that, in this case, | is isomorphic to the group algebra of W; however, in general, this isomorphism cannot be defined without introducing a square root of q (see [1]) It is therefore, natural to extend the ground ring of ~ as follows For any Coxeter group (W, S) we define the Hecke algebra ~ to be J{' | A, where A is the ring of Laurent polynomials with integral coefficients in the indeterminate ql/2 Our purpose is to construct representations oL,Uf endowed with a special basis They will be defined in terms of certain graphs We define a W-graph to be a set of vertices X, with a set Y of edges (an edge is a subset of X consisting of two elements) together with two additional data: for each vertex xeX , we are given a subset I x of S and, for each ordered pair of vertices y, x such that {y, x} e Y, we are given an integer p(y, x) +0 These data are subject to the requirements (10a), (10b) below Let E be

Representations of Coxeter Groups and Hecke Algebras.

Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 527 that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain cate- gories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: sasha@math.mit.edu Department of Mathematics, The University of Chicago, Chicago, Illinois 60637 E-mail address: ginzburg@math.uchicago.edu Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strase 26, D-53 Bonn 3, Germany Current address: Mathematisches Institut, Universitat Freiburg, Albertstrase 23b, D-79104 Freiburg, Germany E-mail address: soergel@sun1.mathematik.uni-freiburg.de License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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Koszul Duality Patterns in Representation Theory

Groupes et algébres de lie

Introduction to lie algebras and representation theory

Representations of Real Reductive Lie Groups

PrefacePrerequisites by ChapterStandard NotationIntroductionIHecke AlgebrasIIThe Category C(g, K)IIIDuality TheoremIVReductive PairsVCohomological InductionVISignature TheoremVIITranslation FunctorsVIIIIrreducibility TheoremIXUnitarizability TheoremXMinimal K TypesXITransfer TheoremXIIEpilog: Weakly Unipotent RepresentationsApp. A. Miscellaneous AlgebraApp. B. Distributions on ManifoldsApp. C. Elementary Homological AlgebraApp. D. Spectral SequencesNotesReferencesIndex of NotationIndex

David A. Vogan

Papers

Representations of Real Reductive Lie Groups

Cohomological Induction and Unitary Representations

Unitary representations with non-zero cohomology

Unitarizability of certain series of representations

Gelfand-Kirillov Dimension for Harish-Chandra Modules.