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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Zonal spherical function ： ウィキペディア英語版 Zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vector in an irreducible representation of ''G''. The key examples are the matrix coefficients of the ''spherical principal series'', the irreducible representations appearing in the decomposition of the unitary representation of ''G'' on ''L''2(''G''/''K''). In this case the commutant of ''G'' is generated by the algebra of biinvariant functions on ''G'' with respect to ''K'' acting by right convolution. It is commutative if in addition ''G''/''K'' is a symmetric space, for example when ''G'' is a connected semisimple Lie group with finite centre and ''K'' is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C * algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant ''L''1 functions is larger; when ''G'' is a semisimple Lie group with maximal compact subgroup ''K'', additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series. Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because ''G'' is the complexification of ''K'', and the formulas are related to analytic continuations of the Weyl character formula on ''K''. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group ''G'' also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of ''G'' on ''L''2(''G''/''K''), as differential operators on the symmetric space ''G''/''K''. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinary differential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function. The name "zonal spherical function" comes from the case when ''G'' is SO(3,R) acting on a 2-sphere and ''K'' is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis. ==Definitions== Let ''G'' be a locally compact unimodular topological group and ''K'' a compact subgroup and let ''H''1 = ''L''2(''G''/''K''). Thus ''H''1 admits a unitary representation π of ''G'' by left translation. This is a subrepresentation of the regular representation, since if ''H''= ''L''2(''G'') with left and right regular representations λ and ρ of ''G'' and ''P'' is the orthogonal projection :$P\; =\backslash int\_K\; \backslash rho(k)\; \backslash ,\; dk$ from ''H'' to ''H''1 then ''H''1 can naturally be identified with ''PH'' with the action of ''G'' given by the restriction of λ. On the other hand, by von Neumann's commutation theorem〔, Algèbres hilbertiennes.〕 :$\backslash lambda(G)^\backslash prime=\; \backslash rho(G)^,$ where ''S denotes the commutant of a set of operators ''S'', so that :$\backslash pi(G)^\backslash prime\; =\; P\; \backslash rho(G)^P.$ Thus the commutant of π is generated as a von Neumann algebra by operators :$P\backslash rho(f)P\; =\; \backslash int\_G\; f(g)\; (P\; \backslash rho(g)P)\; \backslash ,\; dg$ where ''f'' is a continuous function of compact support on ''G''.〔If σ is a unitary representation of ''G'', then $\backslash sigma(f)=\backslash int\_G\; f(g)\backslash sigma(g)\backslash ,\; dg$.〕 However ''P''ρ(''f'') ''P'' is just the restriction of ρ(''F'') to ''H''1, where :$F(g)\; =\backslash int\_K\; \backslash int\_K\; f(kgk^\backslash prime)\; \backslash ,\; dk\backslash ,\; dk^\backslash prime$ is the ''K''-biinvariant continuous function of compact support obtained by averaging ''f'' by ''K'' on both sides. Thus the commutant of π is generated by the restriction of the operators ρ(''F'') with ''F'' in ''C''c(''K''\''G''/''K''), the ''K''-biinvariant continuous functions of compact support on ''G''. These functions form a * algebra under convolution with involution :$F^$ *(g) =\overline, often called the Hecke algebra for the pair (''G'', ''K''). Let ''A''(''K''\''G''/''K'') denote the C * algebra generated by the operators ρ(''F'') on ''H''1. The pair (''G'', ''K'') is said to be a Gelfand pair if one, and hence all, of the following algebras are commutative: * $\backslash pi(G)^\backslash prime$ *$C\_c(K\backslash backslash\; G\; /K)$ *$A(K\backslash backslash\; G\; /K).$ Since ''A''(''K''\''G''/''K'') is a commutative C * algebra, by the Gelfand–Naimark theorem it has the form ''C''0(''X''), where ''X'' is the locally compact space of norm continuous * homomorphisms of ''A''(''K''\''G''/''K'') into C. A concrete realization of the * homomorphisms in ''X'' as ''K''-biinvariant uniformly bounded functions on ''G'' is obtained as follows.〔 Because of the estimate :$\backslash |\backslash pi(F)\backslash |\backslash le\; \backslash int\_G\; |F(g)|\; \backslash ,\; dg\; \backslash equiv\; \backslash |F\backslash |\_1,$ the representation π of ''C''c(''K''\''G''/''K'') in ''A''(''K''\''G''/''K'') extends by continuity to L1(''K''\''G''/''K''), the * algebra of ''K''-biinvariant integrable functions. The image forms a dense * subalgebra of ''A''(''K''\''G''/''K''). The restriction of a * homomorphism χ continuous for the operator norm is also continuous for the norm ||·||1. Since the Banach space dual of L1 is L∞, it follows that :$\backslash chi(\backslash pi(F))\; =\backslash int\_G\; F(g)\; h(g)\; \backslash ,\; dg,$ for some unique uniformly bounded ''K''-biinvariant function ''h'' on ''G''. These functions ''h'' are exactly the zonal spherical functions for the pair (''G'', ''K''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zonal spherical function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.