homogeneous zonal distributions
Transcription
homogeneous zonal distributions
Convolution on homogeneous spaces Capelle, Johan IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1996 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Capelle, J. (1996). Convolution on homogeneous spaces Groningen: s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. 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Paris: Hermann, 1965. 246 Glossary of Symbols > < Italics indicate that the symbol is not in the < long list of fixed notations in Harmonic Analysis a~ a® A acting bilaterally character of A 140 Lie algebra of A convolution) operators in the distributions B normalizer of equal to MAN (.␣|␣.) ´æ(≈) ´æ(≈ )␣%␣␣␣ 139 algebra of G–invariant (or å ‹ ©␣≈ Ì H sesquilinear 75 duality brackets, bilinear 73 quotient of modular functions 117 compactly supported ç°–functions on a manifold ≈ ∂æ(≈) dual of ´(≈) H 31 zonal distributions with inversely 91 Â* 202 diffeomorphisms 106 group of G–invariant % usually denotes involution of zonal sets anti-duality brackets, sesquilinear 75 çµ ∂(≈) 40 compactly supported distributions, character of 108 140 132 80 ~ 78, 92 128 compact supports inner product in a Hilbert space, <.␣,␣.> <.␣|␣.> <␣␣␣␣ smooth functions on ≈ 140, 165, 196 139, 193 a in H >: stabilizer of p in G > <␣␣␣␣ ´(≈) ␣special character of A > in G : extended group G*Ì, 20 or of zonal distributions Ó usually denotes a reproducing Ó° Ó–° operator 35 ¥ 67 ∂¥(≈) distributions on ≈ concentrated on ç°– vectors co–ç°–vectors Hilb˚µ˚˚∂æ(≈) G distributions on the manifold ≈, dual of ∂(≈) 93 89, 92 75 174 174 cone of †µ–invariant ␣ ␣Hilbert G subspaces of ∂æ(≈) 1 76 identity map, or identity character æ ∂æ(≈)H zonal (i.e.: or H–invariant) distributions, on ≈=G/H ∂æ(™)~,␣N * 80 homogeneous zonal distributions ≈ Ω–homogeneous zonal distributions 213 indicator function of the set A 232 K maximal compact subgroup 139, 193 k 139 Lie algebra of K left and right regular translations in the distributions on a group 209 ] * ÈA L,R ≈ ∂æ(™)[¬,N 1,185 LG␣“E¡,E™‘ intertwining space ¬ symmetrizer map Ò␣≈ family of homogeneous 65 74 13, 65 ~ distributions 207 247 — Glossary of Symbols — M M* fl MAp M centralizer of A in K normalizer of A in K unitary dual of M realization of the group submanifold of µ␣␣␣␣␣␣␣␣␣␣␣␣␣ © 140, 193 G/N 140, 194 ≈ ˜ µ µ 148, 152 MA as tu ¤™ 141 operator, in a space of distributions, of multiplication by the function ˜~ †g †xU © space of matrix coefficients 153 the two-point group ”±1’ 73 zonal 83 transpose of u 84 equals †gU␣, when x=gp; ambiguity resolved by U being (complete) Hilbert tensor product of two Hilbert spaces 61, 165 associated to ≈¤a~¤1 fl unitarized version of †g 185 V¤∂ 1¤T ͤTº Ÿ 201 202 202 usually denotes the anti-linear relatively invariant measure on N ì N involution in the zonal distributions, ≈=G/H 72 maximal nilpotent subgroup139, 193 nilpotent group opposite to N ® ®µ ®Ì # Í µ ßs * % Ë(g) Lie algebra of N 139, 196 special weight 139, 196 algebra g ≈ 203 generalized Verma module 148 140 modular function 117 ≈ usually denotes a homogeneous 80 element group, or of Â* 185, 197 Sobolev space 228, 229 in W␣ *␣V 83 218, 219 : push-forward by ^ in ^* : pull-back by ^ in Â* :  without the origin in Ó* : anti-dual of Ó in h* : adjoint of h 179 unitarized push-forward 101 55 56 75 75 operation of g∑G in the distributions on ≈ space ÛT signum character of either a two- 73 13 unitarized right action of Â* is usually the Weyl group n 248 √~ W in ^* * †g v© 73 convolution product 127 universal algebra over the modular function Ÿ û U complexification of the real Lie U␣ ß␣ Ÿ=ì U 141, 196 n 191 Ω spectrum of the distribution T 228 algebra of central convolution operators in ∂æ(≈) 134 INDEX Italics indicate that the term or its particular use is specific to the thesis bi-invariant 131 bilaterally invariant Hilbert subspace 128 convolution operator 128 Bruhat Lemma (decomposition) 140 theory 169, 238n. central compactly supported zonal distributions 134 convolution operators 134 contragredient 72 convolution operator 83 &c. algebra 19-20, 31, 88, 92, 134 ° co–ç –vector 174, 215 direct integral 76, 121, 129, 169, 170, 190 distribution vector 215 effective (action) 100 exchange property 102, 104, 183 fundamental solution 112-116, 159-164, 186187 Generalized Gelfand Pair 77, 105, 131, 172, 188 Hermitian character 135 symmetry 98 Hilbert subspace 74, &c. homogeneous distribution 205-211 space 72 &c induced convolution operators 110-112, 120, 147 Hilbert subspace 120ff., 185 module 10-11, 30-31 inverselybounded or compact 91 Kelvin-inversion 183 kernel (theorem) 80 lattice cone 189, 223 Malgrange-Ehrenpreis 113, 160 multiplicity free decomposition 129, 130, 170 representation 77, 172 Plancherel decomposition and measure 170, 190, 223 of positive type: operators 75, 116-117 zonal distributions 116-117, 237 principal fibre bundle 108 propagator83 pull-back (of a distribution) 79, 101, 195 push-forward (of a distribution) 55, 63, 85 relatively invariant measure 73 reproducing operator 75 Schur’s Lemma (for closeable operators) 137 Sobolev space 228 spectrum (of a distribution on K) 228 strongly zonal 128, 133, 156 symmetric (convolution operators) 211 symmetry 103 (Ë(g),B)–module 70, 148 unitarized push-forward 101, 221 representation 73, 123 249 — Index — Verma module 149 weakly symmetric 97-98, 103 Weil formula 81, 118 Weyl group 140, 194 type 202-205 zonal 80, 83, 91 Ω–homogeneous 211 250