These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

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Geometry of 2D topological field theories

A linear algebraic group over an algebraically closed field k is a subgroup of a group GL n (k) of invertible n × n-matrices with entries in k, whose elements are precisely the solutions of a set of polynomial equations in the matrix coordinates. The present article contains a review of the theory of linear algebraic groups.

Linear Algebraic Groups

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 réductifs sur un corps local

© Gauthier-Villars (Editions scientifiques et medicales Elsevier), 1977, tous droits reserves. L’acces aux archives de la revue « Annales scientifiques de l’E.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

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Induced representations of reductive ${\mathfrak {p}}$-adic groups. I

Lectures on Chevalley Groups

Endomorphisms of linear algebraic groups

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

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Representations of algebraic groups.

© Publications mathématiques de l’I.H.É.S., 1965, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Regular elements of semi-simple algebraic groups

Affine algebraic varieties, affine algebraic groups and their orbits.- First Part: Jordan decompositions, unipotent and diagonalizable groups.- Second Part: Quotients and solvable groups.- Reductive and semisimple algebraic groups, regular and subregular elements.

Robert Steinberg

Papers

Lectures on Chevalley Groups

Endomorphisms of linear algebraic groups

Representations of algebraic groups.

Regular elements of semi-simple algebraic groups

Conjugacy Classes in Algebraic Groups