表示论和复几何

preface

chapter 0. introduction

chapter 1. symplectic geometry

1.1. symplectic manifolds

1.2. poisson algebras

1.3. poisson structures arising from noncommutative algebras

1.4. the moment map

1.5. coisotropic subvarieties

1.6. lagrangian families

chapter 2. mosaic

2.1. hilbert's nullste!lensatz

2.2. atone algebraic varieties

2.3. the deformation construction

2.4. c*-actions on a projective variety

2.5. fixed point reduction

2.6. borel-moore homology

2.7. convolution in borel-moore homology

chapter 3. complex semisimple groups

3.1. semisimple lie algebras and flag varieties

3.2. nilpotent cone

3.3. the steinberg variety

3.4. lagrangian construction of the weyl group

3.5. geometric analysis of h（z）-action

3.6. irreducible representations of we 1 groups

3.7. applications of the jacobson-morozov theorem

chapter 4. springer theory for u（sln）

4.1. geometric construction of the enveloping algebra u（sin（c））

4.2. finite-dimensional simple sln（c）-modules

4.3. proof of the main theorem

4.4. stabilization

chapter 5. equivariant k-theory

5.1. equivariant resolutions

5.2. basic k-theoretic constructions

5.3. specialization in equivariant k-theory

5.4. the koszul complex and the thom isomorphism

5.5. cellular fibration lemma

5.6. the k/inneth formula

5.7. projective bundle theorem and beilinson resolution

5.8. the chern character

5.9. the dimension filtration and "devissage"

5.10. the localization theorem

5.11. functoriality

chapter 6. flag varieties, k-theory, and harmonic polynomials

6.1. equivariant k-theory of the flag variety

6.2. equivariant k-theory of the steinberg variety

6.3. harmonic polynomials

6.4. w-harmonic polynomials and flag varieties

6.5. orbital varieties

6.6. the equivariant hilbert polynomial

6.7. kostant's theorem on polynomial rings

chapter 7. hecke algebras and k-theory

7.1. affine weyl groups and hecke algebras

7.2. main theorems

7.3. case q = h deformation argument

7.4. hilbert polynomials and orbital varieties

7.5. the hecke algebra for sl2

7.6. pwof of the main theorem

chapter 8. representations of convolution algebras

8.1. standard modules

8.2. character formula for standard modules

8.3. constructible complexes

8.4. perverse sheaves and the classification theorem

8.5. the contravariant form

8.6. shed-theoretic analysis of the convolution algebra

8.7. projective modules over convolution algebra

8.8. a non-vanishing result

8.9. semi-small maps

bibliography

《表示论和复几何》是一部经典的从集合角度讲述表示论的高等教程。从几何的角度研究表示论，真可谓“千呼万呼始出来”，尤其是自从1980年D-模型和1990年量子群的箭图，此方法显得更为迫切。表示论的发展顺应科学发展趋势，并且都成功地应用于好多领域，如量子群、仿射李群和量子场论。《表示论和复几何》的前半部分是架起李理论标准知识初学者和数学工作者所需要的广阔背景知识之间的桥梁，为后半部分的学习做好充分的准备。