半單群的表示論（第1卷）

內(nèi)容概要

本書是一部經(jīng)典的著作，分為上下兩卷，前十章為上卷，后六章為下卷。書中講述半單李群表示理論的方式給出了本科目的精華，符合學(xué)習(xí)的自然規(guī)律。定理陳述地相當(dāng)詳細(xì)，增加了許多經(jīng)典的解釋性例子。本章末都有習(xí)題，對于學(xué)習(xí)研究生和科研工作者相當(dāng)有用。目次：理論概述；su(2)，su(2,r)和su(2,c)表示論；c∞向量和通用包絡(luò)代數(shù)；緊李群表示論；非緊群的理論；全純離散系列；導(dǎo)出表示論；可允許表示論；離散系列的結(jié)構(gòu)；全局性質(zhì)；plancherel公式；不可約表示論；最小k型；酉表示；附錄：李群的基本理論；偏微分方程的常規(guī)奇異點；經(jīng)典群的根和受限根。

書籍目錄

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. abstract theory for compact groups  6.application of the abstract 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.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

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