半单群的表示论.第1卷(英文版)

.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.abstract 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.hirai's 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.schrnid's 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-chandra's 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