实半单群的表示理论及相关代数结构

摘要

李群的无限维表示及其相关课题的研究是数学的最活跃的领域之一.该文研究了此领域的一些有趣的问题.该文包括三个部分:1)拟可裂李群的fine表示,好根和R群的分类;2)最小K-型正则的酉表示的Dirac上同调;3)二次李代数.我们回顾了表示理论的发展历史及研究现状,解释了我们感兴趣的问题,并简要说明了我们的研究思路和主要结果.

目录

文摘

英文文摘

Acknowledgements

Introduction

1 Preliminaries

1.1 Notation

1.2 Representation Theory of Real Reductive Groups

1.2.1 Basic Facts

1.2.2 Parabolic Induction

1.2.3 Cohomological Induction

1.2.4 Criteria for Irreducibility and Unitaribility of Representations

2 Fine Representations, Good Roots and R Groups

2.1 Introduction

2.2 Classification of Real Simple Lie algebras: Yen's Results

2.3 Fine Representations and the Set of Good Roots

2.4 The classification of R-groups

2.4.1 Simple split groups

2.4.2 Simple quasisplit (non-split) groups

2.5 Fine K-types

3 The Structure of L: Regular Lowest K-type Case

3.1 Basic Facts

3.2 The Structure of

3.2.1 Main Theorem

3.2.2 Type AⅢ

3.2.3 Types Bl and Cl

3.2.4 Type Dl

4 Dirac Cohomology of Unitary Representations with Regular Lambda-lowest K-types

4.1 Preliminary

4.1.1 Spinors

4.1.2 The Dirac Operator and Dirac Cohomology

4.1.3 (g, K) Cohomology

4.2 Dirac Cohomology of Unitary Representations

4.2.1 The Dominance of μ - ρn

4.2.2 The Dominance of A

4.2.3 The Representations of L

4.2.4 Main Theorem

5 Some Results on Quadratic Lie Algebras

5.1 Introduction

5.2 Preliminary

5.3 The Uniqueness of the Decompositions of Quadratic Lie Algebras

5.4 The Structure of the Elements in 1 and 2

5.5 The Construction of Some Elements in 2

Bibliography