Kac-Moody Groups, their Flag Varieties and Representation Theory

This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in math and math physics, with connections to number theory, combinatorics, topology, singularities, quantum groups, and completely integrable systems. A self-contained text, systematically presented from the Lie algebra to the broader K-M Lie group setting. No prior knowledge of K-M Lie algebras is required. Exposition features numerous examples, illustrations, challenging problems, exercises, and is suitable for grad courses. Also of interest to researchers at the crossroads of representation theory, geometry, and topology.

Klappentext

"Most of these topics appear here for the first time in book form. Many of them are interesting even in the classical case of semi-simple algebraic groups. Some appendices recall useful results from other areas, so the work may be considered self-contained, although some familiarity with semi-simple Lie algebras or algebraic groups is helpful. It is clear that this book is a valuable reference for all those interested in flag varieties and representation theory in the semi-simple or Kac-Moody case."

-MATHEMATICAL REVIEWS

"A lot of different topics are treated in this monumental work. . . . many of the topics of the book will be useful for those only interested in the finite-dimensional case. The book is self contained, but is on the level of advanced graduate students. . . . For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. "

-ZENTRALBLATT MATH

Inhalt
I. Kac-Moody Algebras: Basic Theory.- 1. Definition of Kac-Moody Algebras.- 2. Root Space Decomposition.- 3. Weyl Groups Associated to Kac-Moody Algebras.- 4. Dominant Chamber and Tits Cone.- 5. Invariant Bilinear Form and the Casimir Operator.- II. Representation Theory of Kac-Moody Algebras.- 1. Category $$\mathcal{O}$$.- 2. Weyl-Kac Character Formula.- 3. Shapovalov Bilinear Form.- III. Lie Algebra Homology and Cohomology.- 1. Basic Definitions and Elementary Properties.- 2. Lie Algebra Homology of n-: Results of Kostant-Garland-Lepowsky.- 3. Decomposition of the Category $$\mathcal{O}$$ and some Ext Vanishing Results.- 4. Laplacian Calculation.- IV. An Introduction to ind-Varieties and pro-Groups.- 1. Ind-Varieties: Basic Definitions.- 2. Ind-Groups and their Lie Algebras.- 3. Smoothness of ind-Varieties.- 4. An Introduction to pro-Groups and pro-Lie Algebras.- V. Tits Systems: Basic Theory.- 1. An Introduction to Tits Systems.- 2. Refined Tits Systems.- VI. Kac-Moody Groups: Basic Theory.- 1. Definition of Kac-Moody Groups and Parabolic Subgroups.- 2. Representations of Kac-Moody Groups.- VII. Generalized Flag Varieties of Kac-Moody Groups.- 1. Generalized Flag Varieties: Ind-Variety Structure.- 2. Line Bundles on $${\mathcal{X}^Y}$$.- 3. Study of the Group $${\mathcal{U}^ - }$$.- 4. Study of the Group $${\mathcal{G}^{\min }}$$ Defined by Kac-Peterson.- VIII. Demazure and Weyl-Kac Character Formulas.- 1. Cohomology of Certain Line Bundles on $${Z_\mathfrak{w}}$$.- 2. Normality of Schubert Varieties and the Demazure Character Formula.- 3. Extension of the Weyl-Kac Character Formula and the Borel-Weil-Bott Theorem.- IX. BGG and Kempf Resolutions.- 1. BGG Resolution: Algebraic Proof in the Symmetrizable Case.- 2. A Combinatorial Description of the BGG Resolution.- 3.Kempf Resolution.- X. Defining Equations of $$\mathcal{G}/\mathcal{P}$$ and Conjugacy Theorems.- 1. Quadratic Generation of Defining Ideals of $$\mathcal{G}/\mathcal{P}$$ in Projective Embeddings.- 2. Conjugacy Theorems for Lie Algebras.- 3. Conjugacy Theorems for Groups.- XI. Topology of Kac-Moody Groups and Their Flag Varieties.- 1. The Nil-Hecke Ring.- 2. Determination of $$\bar R$$.- 3. T-equivariant Cohomology of $$\mathcal{G}/\mathcal{P}$$.- 4. Positivity of the Cup Product in the Cohomology of Flag Varieties.- 5. Degeneracy of the Leray-Serre Spectral Sequence for the Fibration $${\mathcal{G}^{\min }} \to {\mathcal{G}^{\min }}/T$$.- XII. Smoothness and Rational Smoothness of Schubert Varieties.- 1. Singular Locus of Schubert Varieties.- 2. Rational Smoothness of Schubert Varieties.- XIII. An Introduction to Affine Kac-Moody Lie Algebras and Groups.- 1. Affine Kac-Moody Lie Algebras.- 2. Affine Kac-Moody Groups.- Appendix A. Results from Algebraic Geometry.- Appendix B. Local Cohomology.- Appendix C. Results from Topology.- Appendix D. Relative Homological Algebra.- Appendix E. An Introduction to Spectral Sequences.- Index of Notation.