Varadarajan V.S. — Introduction to Harmonic Analysis on Semisimple Lie Groups :: Электронная библиотека попечительского совета мехмата МГУ

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Varadarajan V.S. — Introduction to Harmonic Analysis on Semisimple Lie Groups

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Название: Introduction to Harmonic Analysis on Semisimple Lie Groups

Автор: Varadarajan V.S.

Аннотация:

Now in paperback, this graduate-level textbook is an excellent introduction to the representation theory of semi-simple Lie groups. Professor Varadarajan emphasizes the development of central themes in the context of special examples. He begins with an account of compact groups and discusses the Harish-Chandra modules of SL(2,R) and SL(2,C). Subsequent chapters introduce the Plancherel formula and Schwartz spaces, and show how these lead to the Harish-Chandra theory of Eisenstein integrals. The final sections consider the irreducible characters of semi-simple Lie groups, and include explicit calculations of SL(2,R). The book concludes with appendices sketching some basic topics and with a comprehensive guide to further reading. This superb volume is highly suitable for students in algebra and analysis, and for mathematicians requiring a readable account of the topic.

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Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1989

Количество страниц: 316

Добавлена в каталог: 02.04.2005

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Предметный указатель

$\Omega$ -finite eigendistributions on SL(2, R)      208
$\Sigma$       230
$\Xi$       27
(g, K)-modules      112
(g, K)-modules, finite multiplicity theorem for      114
(g, K)-modules, Harish — Chandra finiteness theorem for      113
(g, K)-modules, infinitesimal character of      113
(g, K)-modules, infinitesimal equivalence of      122
(g, K)-modules, unitary      121
Absolutely convergent series in Frechet space      279
Acceptable groups      39
Adjoint representation of a Lie algebra      304
Adjoint representation of a Lie group      303
Analytic vectors for a representation      107
Analytic vectors for a representation, Harish — Chandra density theorem for      109 140—143
Annihilator      11
Asymptotics of matrix elements of SL(2, R)      239
Asymptotics, relation to Plancherel measure of      226 251
Borel section      59
Borel subgroup      68
Cambpell — Baker — Hausdorff formula      306
Cartan subgourp for SL(n)      77
Cartan — Killing classification of simple Lie algebras over C      308
Character formula for discrete series representation of SL(2, R)      200
Character formula for principal and supplementary series representation      73
Character formula for principal series representation of SL(n)      96
Character formula of SL(2, C)      73
Character of       5
Character of       6
Character of a locally compact abelian group      10
Character of finite-dimensional representations      18
Character of irreducible representation, as a distribution      128
Character of irreducible representation, relation to infinitesimal equivalence      129
Character of principal series representations      96
Character of principal series representations, as Fourier transforms of orbital intergals      93
Class function      18
Cocycles      58
Complete reducibility of rational representations of SL(n, C)      49
Complete reducibility of representations of a compact group      20
Completeness theorem for $\mathscr{L}(SL(2,R))$       281
Completeness theorem for $\mathscr{L}_{min}$       272
Completeness theorem for compact groups      21
Cusp forms      282
Degenerate series of representations      71
Differentiable vectors for a representation      106
Differential operators of Lie groups      101 305
Discrete series representations      188
Discrete series representations of SL(2, r)      189
Discrete series representations, formal degree of      188 192
Discriminant of U(n)      25
Distributions on       296
Distributions on manifolds      298
Distributions with unipotent supports on SL(2, R)      217
Distributions with unipotent supports on SL(2, R), transversal order of      218
Distributions, differentiation of      297
Distributions, supports of      298
Duality theorem of Pontrjagin — van Kampen      11
Duel group of a locally compact abelian group      10
Duel measure to Haar measure on a locally compact abelian group      12
Eigendistributions of SL(2, R)      208
Factor      299
Factor representation of a locally compact group      65
Flag      69
Flag of type       69
Flag, full      69
Flag, full isotropic      70
Flag, isotropic      70
Formal adjoint on the universal enveloping algebra      120
Formal transpose of differential operators      307
Formal transpose on the universal enveloping algebra      120
Fourier series of a differentiable vector      109
Fourier series of a function on a compact group      23
Fourier transform of a function of       6
Fourier transform of a function on       5
Fourier transform of a tempered distribution on       9 299
Frechet space      291
Frobenius reciprocity      62
Functional equation of type (m, n)      225 244
Functions of type (m, n) on SL(2, R)      221
Fundamental theorem of Lie      304
Haar measure      301
Haar measure, finiteness theorem of, for (g, K)-modules      113
Haar measure, left      301
Haar measure, matching conditions of, for invatiant eigendistributions on SL(2, R)      214
Haar measure, regulatity theorem of invatiant eigendistributions on SL(2, R)      215—220
Haar measure, right      301
Haar measure, subquotient theorem      118
Harish — Chandra module      115
Harish — Chandra transform of a tempered distribution on SL(2, R)      285
Harish — Chandra transform on $\mathscr{L}(SL(2, R))$       277
Harish — Chandra transform on $\mathscr{L}_{min}$       242
Harish — Chandra, density theorem of, for analytic vectors      109
Highest weight of a representation of a compact Lie group      30
Highest weight of a representation of a semisimple Lie algebra      46
Highest weight, integrality condition fo      47
Imprimitivity theorem      60
Induced representation      58
Infinitesimal character      113
Integrable representations of SL(2, R)      193
Integration formula for Lie algebra of SL(n, C)      161
Integration formula for SL(2, H)      175
Integration formula for SL(n)      95
Integration on manifolds      306
Invariant Harish — Chandra transform of invariant tempered distribution on SL(2, R)      289
Invariant Harish — Chandra transform on Schwartz space of SL(2, R)      285
Invariant Harish — Chandra transform, characterization of      288
Inversion formula for       8
Inversion formula for locally compact abelian groups      11
Iwasawa decomposition for SL(n)      76 81
Jump relations of Harish Chandra for orbital integrals for compactly supported smoth functions on SL(2, R)      185
Jump relations of Harish Chandra for orbital integrals for functions on Schwartz space of SL(2, R)      268
K-finite vectors of a representation      108
Lattice in a locally compact abelian group      11
Lie algebra of a Lie group      302
Lie algebra over a field      302
Limit formula for Lie algebra of SL(n, C)      163
Limit formula for SL(2, R)      185
Limit formula for SL(2, R) on the Schwartz space      268
Limit formula for SL(n, C)      165
Limit formula for U(n)      33
Limits of discrete series of representations of SL(2, R)      201
Locally convex space      291
Maass — Selberg relations      248
Matrix elements of representations of compact groups      19
Matrix elements of representations of Lie group      123

Mellin transform      15
Mock discrete series for SL(2, R)      193
Mumford's comjecture      3
Norm topology for operators      299
Operators of Hilbert — Schmidt class      295
Operators of trace class      295
Operators, integral      296
Operators, summable      296
Orbital integral on Lie algebra of SL(2, R)      170 174
Orbital integral on Lie algebra of SL(n C)      159
Orbital integral on Schwartz space      265 267
Orbital integral on SL(2, R)      170 175
Orbital integral on SL(n, C)      165
Orbital integral on U(n)      32
Orthogonality relations for characters of compact groups      19
Orthogonality relations for matrix elements of compact groups      19
Orthogonality relations for matrix elements of discrete series      188
Parabolic induction      67
Parabolic induction, restricted      76
Parabolic subgroup      68
Parabolic subgroup of SL(n, C)      69
Parabolic subgroup, associated to a Cartan subgroup      78
Parabolic subgroup, Langlands decomposition of      69
Peter — Weyl theorem      21
Plancherel formula for       7
Plancherel formula for in distribution form      9
Plancherel formula for       6
Plancherel formula for compact groups      2
Plancherel formula for locally compact abelian groups      12
Plancherel formula for SL(2, R)      205
Plancherel formula for SL(2, R) on Schwartz space      270
Plancherel formula for SL(n, C)      166
Plancherel formula for the unitary group U(n)      35
Plancherel formula, general form of, for semisimple groups      155
Plancherel formula, relation to asymptotics of matrix elements      226 251
Poincare — Birkhoff — Witt theorem      103 305
Poisson summation formula on       10
Poisson summation formula on locally compact abelian group      11
Principal series representations of SL(n)      70 85
Principal series representations of SL(n), characters of      70 96
Principal series representations of SL(n), compact model for      84
Principal series representations of SL(n), irreducibility of, for SL(2, C)      98
Principal series representations of SL(n), irreducibility of, for SL(n)      148
Projection valued measure(pv,)      292
Quasi-invariant measure      56
Quasicharacter      79
Quasiregular representation      57
Radial component of Casimir operator on Lie algebra of SL(2, R)      179
Radial component of Casimir operator on SL(2, R)      180
Rank of a compact Lie group      36
Regular point of a compact Lie group      36
Regular point of U(n)      25
Regular representation of       6
Regular representation of       5
Regular representation of a compact group      20
Representations of a group      1
Representations of a topological group      2
Representations, analytic of a Lie group      42
Representations, complete reducibility of, of a compact group      20
Representations, contragradient      1
Representations, direct sum of      1
Representations, holomorphic, of a complex Lie group      42
Representations, irreducible      1
Representations, irreducible, of SL(2, C)      45
Representations, projective unitary      4
Representations, rational      2
Representations, tensor product of      1
Representations, unitary      2
Representations, unitary equivalence of      2
Resolvent set of an operator      291
Root spaces of Lie algebra of a compact Lie group      36
Roots of Lie algebra of a compact Lie group      36
Roots, global      36
Roots, positive      37
Roots, simple      37
Schwartz space of       7
Schwartz space of SL(2, R)      230
Self-adjoint operator      292 294
Self-adjoint operator, spectral projections of      292
Self-adjoint operator, spectral subspaces of      292
Self-adjoint operator, spectral theorem for      292 294
Semidirect products, regular      64
Semidirect products, representations of      64
Seminarm      291
semisimple Lie algebras      309
Simple Lie algebras      308
Simple Lie algebras, Cartan — Killing classification of      308
Singular set of U(n)      26
Spherical functions on SL(2, R)      221
Strong inequality for functions on SL(2, R)      81 230
Strong topology for operators      299
Supplementary series, representations      71
Supplementary series, representations of SL(2, C)      73 147
Supplementary series, the characters of      73
Symmetric operator      293
Symmetric operator, Cayley transfrorm of      294
Symmetric operator, deficiency indices of      294
System of imprimitivity      60
System of imprimitivity, ergodic      61
Tannaka — Chevalley theorem      51
Tempered distribution on       9 299
Tempered distribution on SL(2, R)      260
Tempered measures on SL(2, R)      260
Type I groups      65
Unimodular group      55 301
Unitarian trick      50
Unitary dual of a group      2
Universal enveloping algebra      101 304
Universal enveloping algebra, filtration on      102
van Neumann algebra      299
Wave packet      240
Wave packet theorem      240
Wave packet, exact      255
Wave packet, normalized      255
Weak inequality for functions on SL(2, R)      230
Weak topology for operators      299
Weight of a representation of U(n)      30
Weight of representation of a semisimple Lie algebra      46
Weyl chamber      37
Weyl group      35
Weyl, character formula of, for a compact Lie group      40
Weyl, character formula of, for U(n)      28
Weyl, dimension formula of, for a compact Lie group      40
Weyl, dimension formula of, for U(n)      32
Weyl, integration formula of, for U(n)      27
Weyl, theorem of, on the finiteness of the universal covering group of a compact semisimple Lie group      41

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