半单群的表示论. 第1卷

preface to the princeton landmarks in mathematics edition

preface

acknowledgments

chapter i. scope of the theory

1.the classical groups

2.cartan decomposition

3.representations

4.concrete problems in representation theory

5. abstractModel theory for compact groups

6.application of the abstractModel theory to lie groups

7.problems

chapter ii. representations of su（2）， sl（2， r）， and sl（2， c）

1.the unitary trick

2.irreducible finite-dimensional complex-linear representations of si（2， c）

3.finite-dimensional representations of s1（2， c）

4.irreducible unitary representations of sl（2， c）

5.irreducible unitary representations of sl（2， r）

6.use of su（1， 1）

7.plancherel formula

8.problems

chapter iii. c∞ vectors and the universal enveloping algebra

1.universal enveloping algebra

2.actions on universal enveloping algebra

3.c∞vectors

4.gatrding subspace

5.problems

chapter iv. representations of compact lie groups

1.examples of root space decompositions

2.roots

3.abstractModel root systems and positivity

4.weyl group， algebraically

5.weights and integral forms

6.centalizers of tori

7.theorem of the highest weight

8.verma modules

9.weyl group， analytically

10.weyl character formula

11.problems

chapter v. structure theory for noncompact groups

1.cartan decomposition and the unitary trick

2.iwasawa decomposition

3.regular elements， weyl chambers， and the weyl group

4.other decompositions

5.parabolic subgroups

6.integral formulas

7.borel-weil theorem

8.problems

chapter vi. holomorphic discrete series

1.holomorphic discrete series for su（1， 1）

2.classical bounded symmetric domains

3.harish-chandra decomposition

4.holomorphic discrete series

5.finiteness of an integral

6.problems

chapter vii. induced representations

1.three pictures

2.elementary properties

3.bruhat theory

4.formal intertwining operators

5.gindikin-karpelevi formula

6.estimates on intertwining operators， part i

7.analytic continuation of intertwining operators， part i

8.spherical functions

9.finite-dimensional representations and the h function

10.estimates on intertwining operators， part ii

11.tempered representations and langlands quotients

12.problems

chapter viii. admissible representations

1.motivation

2.admissible representations

3.invariant subspaces

4.framework for studying matrix coefficients

5.harish-chandra homomorphism

6.infinitesimal character

7.differential equations satisfied by matrix coefficients

8.asymptotic expansions and leading exponents

9.first application： subrepresentation theorem

10.second application： analytic continuation of interwining operators， part ii

11.third application： control of k-finite z（gc）-finite functions

12.asymptotic expansions near the walls

13.fourth application： asymptotic size of matrix coefficients

14.fifth application： identification of irreducible tempered representations

15.sixth application： langlands classification of irreducible admissible representations

16.problems

chapter ix. construction of discrete series

1.infinitesimally unitary representations

2.a third way of treating admissible representations

3.equivalent definitions of discrete series

4.motivation in general and the construction in su（1， 1）

5.finite-dimensional spherical representations

6.duality in the general case

7.construction of discrete series

8.limitations on k types

9.lemma on linear independence

10.problems

chapter x. global characters

1.existence

2.character formulas for sl（2， r）

3.induced characters

4.differential equations

5.analyticity on the regular set， overview and example

6.analyticity on the regular set， general case

7.formula on the regular set

8.behavior on the singular set

9.families of admissible representations

10.problems

chapter xi. introduction to plancherel formula

1.constructive proof for su（2）

2.constructive proof for sl（2， c）

3.constructive proof for sl（2， r）

4.ingredients of proof for general case

5.scheme of proof for general case

6.properties of fi

7.hirais patching conditions

8.problems

chapter xii. exhaustion of discrete series

1.boundedness of numerators of characters

2.use of patching conditions

3.formula for discrete series characters

4.schwartz space

5.exhaustion of discrete series

6.tempered distributions

7.limits of discrete series

8.discrete series of m

9.schrnids identity

10.problems

chapter xiii. plancherel formula

1.ideas and ingredients

2.real-rank-one groups， part i

3.real-rank-one groups， part ii

4.averaged discrete series

5.sp （2， r）

6.general case

7.problems

chapter xiv. irreducible tempered representations

1.sl（2， r） from a more general point of view

2.eisenstein integrals

3.asymptotics of eisenstein integrals

4.the η functions for intertwining operators

5.first irreducibility results

6.normalization of intertwining operators and reducibility

7.connection with plancherel formula when dim a = 1

8.harish-chandras completeness theorem

9.r group

10.action by weyl group on representations of m

11.multiplicity one theorem

12.zuckerman tensoring of induced representations

13.generalized schmid identities

14.inversion of generalized schmid identities

15.complete reduction of induced representations

16.classification

17.revised langlands classification

18.problems

chapter my. minimal k types

1.definition and formula

2.inversion problem

3.connection with intertwining operators

4.problems

chapter xvi. unitary representations

1.sl（2， r） and sl（2， c）

2.continuity arguments and complementary series

3.criterion for unitary representations

4.reduction to real infinitesimal character

5.problems

appendix a： elementary theory of lie groups

1.lie algebras

2.structure theory of lie algebras

3.fundamental group and covering spaces

4.topological groups

5.vector fields and submanifolds

6.lie groups

appendix b： regular singular points of partial differential equations

1.summary of classical one-variable theory

2.uniqueness and analytic continuation of solutions in several variables

3.analog of fundamental matrix

4.regular singularities

5.systems of higher order

6.leading exponents and the analog of the indicial equation

7.uniqueness of representation

appendix c： roots and restricted roots for classical groups

1.complex groups

2.noncompact real groups

3.roots vs. restricted roots in noncompact real groups

notes

references

index of notation

index

《半单群的表示论（第1卷）》是一部经典的著作，分为上下两卷，前十章为上卷，后六章为下卷。书中讲述半单李群表示理论的方式给出了本科目的精华，符合学习的自然规律。定理陈述地相当详细，增加了许多经典的解释性例子。本章末都有习题，对于学习研究生和科研工作者相当有用。目次：理论概述；su（2），su（2，r）和su（2，c）表示论；向量和通用包络代数；紧李群表示论；非紧群的理论；全纯离散系列；导出表示论；可允许表示论；离散系列的结构；全局性质；plancherel公式；不可约表示论；最小k型；酉表示；附录：李群的基本理论；偏微分方程的常规奇异点；经典群的根和受限根。