Kac-Moody Groups, Their Flag Varieties, and Representation Theory (Freiburger Veroffentlichungen Zum Religionsrecht,)

Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge­ bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan­ dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g. Progress in Mathematics 204......Page 1 Kac-Moody Groups, their Flag Varieties and Representation Theory......Page 2 Contents......Page 5 Preface......Page 9 Convention......Page 12 I. Kac-Moody Algebras: Basic Theory......Page 15 1.1. Definition of Kac-Moody Algebras......Page 16 1.2. Root Space Decomposition......Page 19 1.3. Weyl Groups Associated to Kac-Moody Algebras......Page 25 1.4. Dominant Chamber and Tits Cone......Page 41 1.5. Invariant Bilinear Form and the Casimir Operator......Page 43 II. Representation Theory of Kac-Moody Algebras......Page 53 2.1. Category O......Page 54 2.2. Weyl-Kac Character Formula......Page 61 2.3. Shapavalov Bilinear Form......Page 65 III. Lie Algebra Homology and Cohomology......Page 81 3.1. Basic Definitions and Elementary Properties......Page 82 3.2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky......Page 94 3.3. Decomposition of the Category O and some Ext Vanishing Results......Page 104 3.4. Laplacian Calculation......Page 111 IV. An Introduction to ind-Varieties and pro-Groups......Page 122 4.1. Ind-Varieties: Basic Definitions......Page 123 4.2. Ind-Groups and their Lie Algebras......Page 127 4.3. Smoothness of ind-Varieties......Page 135 4.4. An Introduction to pro-Groups and pro-Lie Algebras......Page 142 5.1. An Introduction to Tits Systems......Page 162 5.2. Refined Tits Systems......Page 179 VI. Kac-Moody Groups: Basic Theory......Page 186 6.1. Definition of Kac-Moody Groups and Parabolic Subgroups......Page 187 6.2. Representations of Kac-Moody Groups......Page 200 VII. Generalized Flag Varieties of Kac-Moody Groups......Page 212 7.1. Generalized Flag Varieties„Ind-Variety Structure......Page 214 7.2. Line Bundles on XY......Page 232 7.3. Study of the group U......Page 234 7.4. Study of the Group Gmin Defined by Kac-Peterson......Page 241 VIII. Demazure and Weyl-Kac Character Formulas......Page 258 8.1. Cohomology of Certain Line Bundles on Zw......Page 259 8.2. Normality of Schubert Varieties and the Demazure Character Formula......Page 286 8.3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem......Page 294 IX. BGG and Kempf Resolutions......Page 308 9.1. BGG Resolution: An Algebraic Proof in the Symmetrizable Case......Page 310 9.2. A Combinatorial Description of the BGG Resolution......Page 318 9.3. Kempf Resolution......Page 334 X. Defining Equations of G/P and Conjugacy Theorems......Page 350 10.1. Quadratic Generation of Defining Ideals of G/P in Projective Embeddings......Page 352 10.2. Conjugacy Theorems for Lie Algebras......Page 360 10.3. Conjugacy Theorems for Groups......Page 371 XI. Topology of Kac-Moody Groups and Their Flag Varieties......Page 382 11.1. The Nil-Hecke Ring......Page 384 11.2. Determination of R......Page 405 11.3. T-equivariant Cohomology of G/B......Page 409 11.4. Positivity of the Cup Product in the Cohomology of Flag Varieties......Page 429 þÿ......Page 440 XII. Smoothness and Rational Smoothness of Schubert Varieties......Page 460 12.1. Singular Locus of Schubert Varieties......Page 462 12.2. Rational Smoothness of Schubert Varieties......Page 478 XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups......Page 494 13.1. Affine Kac-Moody Lie Algebras......Page 495 13.2. Affine Kac-Moody Groups......Page 503 Appendix A. Results from Algebraic Geometry......Page 524 Appendix B. Local Cohomology......Page 540 Appendix C. Results from Topology......Page 546 Appendix D. Homological Algebra......Page 552 Appendix E. An Introduction to Spectral Sequences......Page 562 Bibliography......Page 572 Index of Notation......Page 604 Index......Page 610

This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in mathematics and mathematical physics. K-M Lie algebras were introduced in the mid-sixties independently by V. Kac and R. Moody, generalizing finite-dimensional semisimple Lie algebras. K-M theory has since undergone tremendous developments in various directions and has profound connections with a number of diverse areas, including number theory, combinatorics, topology, singularities, quantum groups, completely integrable systems, and mathematical physics.
This comprehensive, well-written text moves from K-M Lie algebras to the broader K-M Lie group setting, and focuses on the study of K-M groups and their flag varieties. In developing K-M theory from scratch, the author systematically leads readers to the forefront of the subject, treating the algebro-geometric, topological, and representation-theoretic aspects of the theory. Most of the material presented here is not available anywhere in the book literature.
{\it Kac—Moody Groups, their Flag Varieties and Representation Theory} is suitable for an advanced graduate course in representation theory, and contains a number of examples, exercises, challenging open problems, comprehensive bibliography, and index. Research mathematicians at the crossroads of representation theory, geometry, and topology will learn a great deal from this text; although the book is devoted to the general K-M case, those primarily interested in the finite-dimensional case will also benefit. No prior knowledge of K-M Lie algebras or of (finite-dimensional) algebraic groups is required, but some basic knowledge would certainly be helpful. For the reader's convenience some of the basic results needed from other areas, including ind-varieties, pro-algebraic groups and pro-Lie algebras, Tits systems, local cohomology, equivariant cohomology, and homological algebra are included.

I. Kac-moody Algebras: Basic Theory -- Ii. Representation Theory Of Kac-moody Algebras -- Iii. Lie Algebra Homology And Cohomology -- Iv. An Introduction To Ind-varieties And Pro-groups -- V. Tits Systems: Basic Theory -- Vi. Kac-moody Groups: Basic Theory -- Vii. Generalized Flag Varieties Of Kac-moody Groups -- Viii. Demazure And Weyl-kac Character Formulas -- Ix. Bgg And Kempf Resolutions -- X. Defining Equations Of G/p And Conjugacy Theorems -- Xi. Topology Of Kac-moody Groups And Their Flag Varieties -- Xii. Smoothness And Rational Smoothness Of Schubert Varieties -- Xiii. An Introduction To Affine Kac-moody Lie Algebras And Groups -- App. A. Results From Algebraic Geometry -- App. B. Local Cohomology. Shrawan Kumar. Includes Bibliographical References (p. [559]-589) And Indexes.