DOCSLIB.ORG

Harish-Chandra's C-Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Harish-Chandra's C-Function ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques SIMON GINDIKIN Harish-Chandra’s c-function; 50 years later Tome XXV, no 2-3 (2016), p. 385-402. <http://afst.cedram.org/item?id=AFST_2016_6_25_2-3_385_0> © Université Paul Sabatier, Toulouse, 2016, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XXV, n◦ 2-3, 2016 pp. 385-402 Harish-Chandra’s c-function; 50 years later Simon Gindikin(1) RESUM´ E.´ — Nous discutons diff´erents aspects de la fonction c de Harish- Chandra, en soulignant ses interactions avec la transform´ee horosph´erique. ABSTRACT. — We discuss different aspects of the c-function of Harish- Chandra with focus on its connection with the horospherical transform. I have known Vadim Schechtman for many years, but it was only a few years ago that I found out that we share two strong interests: sur- prises which it is possible yet to mine in works of great mathematicians and product-formulas for some special functions on Lie groups. Vadim has col- lected a very impressive exhibition of such formulas (see unpublished notes on his webpage). It starts with the product-formula for the c-function which Karpelevich and I found more than 50 years ago [19] in the beginning of my mathematical life. Vadim’s anniversary for me is a pleasant reason to talk about this old subject. Starting at least with Gauss (who continuously pro- duced new proofs of the Fundamental Theorem of Algebra), old mathemati- cians like to return to subjects of their first mathematical love. Perhaps it has a similar nature to criminals, returning to the scene of the crime? What about the c-function, for the past several years, there were quite a few pop- ular generalizations of the product-formula on arbitrary fields, but in these notes we will stay inside old fashioned real or complex considerations. (1) Departm. of Math., Hill Center, Rutgers University, 110 Frelinghysen Road, Pis- cataway, NJ 08854 gindikinath.rutgers.edu – 385 – Simon Gindikin Origin of the c-function The c-function appeared in the publication of Harish-Chandra of 1958 [22] about the asymptotic of zonal spherical functions on Riemannian sym- metric spaces of non compact type and its applications to the Plancherel formula on these spaces. So it appeared as an important, but intermediate object. Let us start from a discussion if it has an independent conceptual nature. Often such kind of questions are artificial, but I believe that here it is a perfectly legitimate. Let us remind of the definitions. Let G be a simple connected semisimple Lie group; K be its maxi- mal compact subgroup; A, N be complimentary Cartanian and maximal unipotent subgroups; G = KAN be the Iwasawa decomposition; N¯ be the unipotent subgroup opposite to N and X = G/K be the Riemannian symmetric space. Let a(g) be the projection on the factor A at the Iwasawa decomposition, a be the Lie algebra Lie of A, a∗ be the dual space, Σ, Π be the system of positive and simple roots, ρ be the half-sum of positive roots; aξ = exp(ξ,log a),ξ a, a A. ∈ ∈ We can extend this definition on ξ Ca. Let μ(dn¯) be the invariant measure on N¯. Now we can define the c-function∈ ρ iλ c(λ)= a(¯n)− − μ(dn¯). ¯ N It is possible to prove that this integral is absolutely convergent if Re(iλ) lies in the positive Weyl chamber W+. Let M be the centralizer of A at K and we call Ξ=G/MN¯ the horospherical space. Points of Ξ parameterize non degenerate orbits E(ξ) of unipotent subgroups conjugated to N which are called the horospheres. On Ξ we have a “left” action of A commutating with the action G, since A normalizes MN¯. Then the horospherical space Ξ fibers on A-fibers over the flag manifold F =Ξ/A = G/AMN.¯ So the isotropy subgroup of F is parabolic. – 386 – Harish-Chandra’s c-function; 50 years later In E.Cartan’s conception the irreducible finite dimensional spherical rep- resentations on X are characterized by unique K-invariant elements - zonal spherical functions. Another possibility is to connect with them N-invariant elements - highest weight vectors. This duality for spherical representations extends from finite dimensional representations on some infinite dimensional representations. It requires some analytical justifications which we will not discuss here. At Borel-Weil modification of highest weight method the irreducible rep- resentations are realized at sections of line bundles on the flag spaces F . Gelfand-Graev’s conception of integral geometry suggests realizing of ir- reducible representations at functions on the horospherical space Ξ. If we decompose the representation at functions on Ξ relative to the “left” ac- tion of A we will obtain Borel-Weil realizations of irreducible components on F , which are parameterized by characters of A. This looks as a small modification, but since points of Ξ admit a geometrical realization at X, it opens a possibility for a new construction of geometric analysis. Gelfand and Graev considered the integral geometry corresponding to infinite di- mensional representations [6]. We will discuss later a similar approach to finite dimensional representations. So irreducible spherical representations can be realized either on X or on Ξ. The principal moment of Gelfand-Graev’s approach was that since spectrums on X and Ξ coincide it must be an invariant intertwining oper- ator between functions on X and Ξ. This is Radon’s type operator - the integration along horospheres - the horospherical transform: f(ξ)= f(x)μ(dn),ξ Ξ, H ∈ E(ξ) where μ(dn) is the invariant measure on N¯ translated on the horospheres. The first area of applications was the Plancherel formula on complex semisim- ple Lie groups and some homogeneous spaces: since the decomposition on irreducible ones is reduced to commutative Fourier transform on A, we will have the decomposition as soon as we can invert the horospherical trans- form. Later we will return to this possibility. Heuristically, the horospherical transform is scalar on irreducible con- stituents if we identify them on X and Ξ. Modulo all complications con- nected with the continuous spectrum and non L2 eigenfunctions the c- function represents the eigenvalues of the horospherical transform under a natural normalization. I believe that this heuristic view is crucial for the understanding of the nature of the c-function. Its appearance at the asymp- totic of the zonal spherical function is already a secondary event. – 387 – Simon Gindikin As test functions for the definition of eigenvalues let us consider the “highest weight vectors” on X and Ξ: ρ iλ ρ iλ χ(x λ)=a(x)− − ,x X; χ(ξ λ)=a(ξ)− − ,ξ Ξ; | ∈ | ∈ Here a(ξ) is the projection on A in the decomposition NMAN¯ at the open part of G and, as a consequence, of Ξ. Then χ(ξ λ) can be interpreted as boundary values of χ(x λ). | | The direct computation shows that χ(x λ)=c( λ)χ(ξ λ) H | − | if Re( iλ) is in the positive Weyl chamber W+ and we take boundary values for real− λ. There is a natural intertwining operator acting from Ξ to X - the dual horospherical transform F (x)= F (ξ)μ(dξ). P S(x) Here S(x) is the set of parameters of horospheres passing through x X. We have S(x)=K/M and μ(dξ) is the invariant measure. So in the∈ dual horospherical transform we have the integration on the compact manifold K/M. Let us remark that χ(ξ λ) is defined only on an open set of Ξ and | the integral converges only for Re(iλ) W+. On the highest weight vectors it is scalar: ∈ χ(ξ λ)=d(λ)χ(x λ). P | | The direct computation shows that d(λ)=c(λ),Re(iλ) W . ∈ + We need just to apply the Harish-Chandra’s trick of the replacement of integrations along compact orbits by integrations along unipotent ones [23]. He applied it for the integral representation of zonal spherical functions on the unipotent subgroup. It corresponds to the appearance of the factor c(λ) at the Poisson formula for reconstructing eigenfunctions of invariant differential operators on X through the boundary values on the boundary F ([23, 24] . The integrals defining the direct and dual horospherical transforms , have no joint convergence area, but we can in both cases consider the bound-P H ary values for real λ and then their composition will have the eigenvalues c(λ)c( λ)= c(λ) 2. − | | – 388 – Harish-Chandra’s c-function; 50 years later So if we add the operator with the eigenvalues L p(λ)=1/ c(λ) 2 | | we have the identical operator and the operator inverts . It illus- trates the principal Harish-Chandra’sLPH observation thatLPp(λ) is theH density in the Plancherel formula and is the reason why in the Plancherel density appears the square of modulus of the c-function. It would be interesting to transform these heuristic discussions into rig- orous statements.
Load more

Recommended publications
  • Q-KRAWTCHOUK POLYNOMIALS AS SPHERICAL FUNCTIONS on the HECKE ALGEBRA of TYPE B
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 10, Pages 4789{4813 S 0002-9947(00)02588-5 Article electronically published on April 21, 2000 q-KRAWTCHOUK POLYNOMIALS AS SPHERICAL FUNCTIONS ON THE HECKE ALGEBRA OF TYPE B H. T. KOELINK Abstract. The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of q-Krawtchouk polynomials using the quantised enveloping algebra for sl(2; C). The result covers a number of previously established interpretations of (q-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum SU(2) group. 1. Introduction The hyperoctahedral group is the finite group of signed permutations, and it con- tains the permutation group as a subgroup. The functions on the hyperoctahedral group, which are left and right invariant with respect to the permutation group, are spherical functions. The zonal spherical functions are the spherical functions which are contained in an irreducible subrepresentation of the group algebra under the left regular representation. The zonal spherical functions are known in terms of finite discrete orthogonal polynomials, the so-called symmetric Krawtchouk poly- nomials. This result goes back to Vere-Jones in the beginning of the seventies, and is also contained in the work of Delsarte on coding theory; see [9] for information and references. There are also group theoretic interpretations of q-Krawtchouk polynomials. Here q-Krawtchouk polynomials are q-analogues of the Krawtchouk polynomials in the sense that for q tending to one we recover the Krawtchouk polynomials.
    [Show full text]
  • Kobe University Repository : Kernel
    Kobe University Repository : Kernel タイトル Zonal spherical functions on the quantum homogeneous space Title SUq(n+1)/SUq(n) 著者 Noumi, Masatoshi / Yamada, Hirofumi / Mimachi, Katsuhisa Author(s) 掲載誌・巻号・ページ Proceedings of the Japan Academy. Ser. A, Mathematical Citation sciences,65(6):169-171 刊行日 1989-06 Issue date 資源タイプ Journal Article / 学術雑誌論文 Resource Type 版区分 publisher Resource Version 権利 Rights DOI 10.3792/pjaa.65.169 JaLCDOI URL http://www.lib.kobe-u.ac.jp/handle_kernel/90003238 PDF issue: 2021-10-01 No. 6] Proc. Japan Acad., 65, Ser. A (1989) 169 Zonal Spherical Functions on the Quantum Homogeneous Space SU (n + 1 )/SU (n) By Masatoshi NOUMI,*) Hirofumi YAMADA,**) and Katsuhisa :M:IMACHI***) (Communicated by Shokichi IYANA,GA, M. ff.A., June. 13, 1989) In this note, we give an explicit expression to the zonal spherical func- tions on the quantum homogeneous space SU(n + 1)/SU(n). Details of the following arguments as well as the representation theory of the quantum group SU(n+ 1) will be presented in our forthcoming paper [3]. Through- out this note, we fix a non-zero real number q. 1. Following [4], we first make a brief review on the definition of the quantum groups SLy(n+ 1; C) and its real form SUb(n+ 1). The coordinate ring A(SL,(n+ 1; C)) of SLy(n+ 1; C) is the C-algebra A=C[x,; O_i, ]_n] defined by the "canonical generators" x, (Oi, ]n) and the following fundamental relations" (1.1) xixj=qxx, xixj=qxx or O_i]_n, O_k_n, (1.2) xx=xxt, xx--qxxj=xtx--q-lxx or Oi]_n, Okl_n and (1.3) det-- 1.
    [Show full text]
  • Arxiv:Math/9512225V1
    SOME REMARKS ON THE CONSTRUCTION OF QUANTUM SYMMETRIC SPACES Mathijs S. Dijkhuizen Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan 31 May 1995 Abstract. We present a general survey of some recent developments regarding the construction of compact quantum symmetric spaces and the analysis of their zonal spherical functions in terms of q-orthogonal polynomials. In particular, we define a one-parameter family of two-sided coideals in Uq(gl(n, C)) and express the zonal spherical functions on the corresponding quantum projective spaces as Askey-Wilson polynomials containing two continuous and one discrete parameter. 0. Introduction In this paper, we discuss some recent progress in the analysis of compact quantum symmetric spaces and their zonal spherical functions. It is our aim to present a more or less coherent overview of a number of quantum analogues of symmetric spaces that have recently been studied. We shall try to emphasize those aspects of the theory that distinguish quantum symmetric spaces from their classical counterparts. It was recognized not long after the introduction of quantum groups by Drinfeld [Dr], Jimbo [J1], and Woronowicz [Wo], that the quantization of symmetric spaces was far from straightforward. We discuss some of the main problems that came up. First of all, there was the, at first sight rather annoying, lack of interesting quan- tum subgroups. Let us mention, for instance, the well-known fact that there is no analogue of SO(n) inside the quantum group SUq(n). A major step forward in overcoming this hurdle was made by Koornwinder [K2], who was able to construct a quantum analogue of the classical 2-sphere SU(2)/SO(2) and express the zonal arXiv:math/9512225v1 [math.QA] 21 Dec 1995 spherical functions as a subclass of Askey-Wilson polynomials by exploiting the notion of a twisted primive element in the quantized universal enveloping algebra q(sl(2, C)).
    [Show full text]
  • Noncommutative Fourier Transform for the Lorentz Group Via the Duflo Map
    PHYSICAL REVIEW D 99, 106005 (2019) Noncommutative Fourier transform for the Lorentz group via the Duflo map Daniele Oriti Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany and Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstrasse 37, D-80333 München, Germany, EU Giacomo Rosati INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy (Received 11 February 2019; published 13 May 2019) We defined a noncommutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the noncommutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map. DOI: 10.1103/PhysRevD.99.106005 I. INTRODUCTION of (noncommutative) Fourier transform to establish the (unitary) equivalence between this new representation and Physical systems whose configuration or phase space is the one based on L2 functions on the group configura- endowed with a curved geometry include, for example, tion space. point particles on curved spacetimes and rotor models in The standard way of dealing with this issue is indeed to condensed matter theory, and they present specific math- renounce to a representation that makes direct use ematical challenges in addition to their physical interest. of the noncommutative Lie algebra variables, and to use In particular, many important results have been obtained in instead a representation in terms of irreducible representa- the case in which their domain spaces can be identified tions of the Lie group, i.e., to resort to spectral decom- with (Abelian) group manifolds or their associated homo- position as the proper analogue of the traditional Fourier geneous spaces.
    [Show full text]
  • Wiener's Tauberian Theorem for Spherical Functions on the Automorphism Group of the Unit Disk
    http://www.diva-portal.org This is the published version of a paper published in Arkiv för matematik. Citation for the original published paper (version of record): Hedenmalm, H., Benyamini, Y., Ben Natan, Y., Weit, Y. (1996) Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk. Arkiv för matematik, 34: 199-224 Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-182412 Ark. Mat., 34 (1996), 199-224 (~) 1996 by Institut Mittag-Leffier. All rights reserved Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk Yaakov Ben Natan, Yoav Benyamini, Hs Hedenmalm and Yitzhak Weit(1) Abstract. Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (con- formal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a "two circles" Morera type theorem (earlier announced by AgranovskiY). Introduction If G is a locally compact abelian group, Wiener's tanberian theorem asserts that if the Fourier transforms of the elements of a closed ideal I of the convolution algebra LI(G) have no common zero, then I=LI(G).
    [Show full text]
  • On a Representation of the Group GL (K; R) and Its Spherical Functions
    Mem Fac ut & scl Shrmane Univ., Nat. Sci. 4, pp. Io 06 March 15 1971 On a RepreSentation of The Group GL (k; R) and ItS Spherical FunctionS Yasuhiro ASOo (Department of Mathematlcs shlmane Unrversity, Mat**ue, Japan) (Received October 12, 1970) In 1961, A. T. James [1] introduced the zonal polynomial of real positive definite matrix and he described some properties and a method of calculation of it. Lately, he [2] also showed that it is ' an eigenfunction of the Laplace-~eltrami operator. The '_onal polynomial plays significant roles in di-~*tribution problem** of eigenvalues related to the normal multivariate distri.bution [3]. In the present paper, we will describe a representation of the group G'L (k ; R) and its spherical functions guided by N. J. Vilenkin [4]. Our assumptions I and 2 in the following may be satisfied with zonal polynomials. We also give a deLinition of (zonal) spherical function of the group GL (k ; R) guided by K. Maurin [5] and we show that our zonal spherical function is in agreement with the latter definition. S I . Definitiola of spherical functions and son~_e properties related to the representation A(2f) (g) of th:e group GL (k ; R) Let { {g}(2)}(f), g E GL (k ; R) be a representation of the group GL (k ; R) on the space Vf of hornogeneous polynomials of degree f of the matrix ~;t ~ S, where ~;t is the space of real positive definite matrices of order k. Owing to Thrall [6]-Hua [71 's result, the representation space space Vf is completely decomposable mto rrreducrble mvanant subspaces V(f) on which representation A(2f,,.
    [Show full text]
  • Arxiv:0809.2017V1 [Math.OC] 11 Sep 2008 Lc Ignlzto,Teafnto,Pstv Kernels, Harmonics
    LECTURE NOTES: SEMIDEFINITE PROGRAMS AND HARMONIC ANALYSIS FRANK VALLENTIN Abstract. Lecture notes for the tutorial at the workshop HPOPT 2008 — 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands. Contents 1. Introduction 2 2. The theta function for distance graphs 3 2.1. Distance graphs 4 2.2. Original formulations for theta 6 2.3. Positive Hilbert-Schmidt kernels 7 2.4. The theta function for infinite graphs 9 2.5. Exploiting symmetry 10 3. Harmonic analysis 12 3.1. Basic tools 13 3.2. The Peter-Weyl theorem 14 3.3. Bochner’s characterization 15 4. Boolean harmonics 16 4.1. Specializations 16 4.2. Linearprogrammingboundforbinarycodes 17 4.3. Fourier analysis on the Hamming cube 18 5. Spherical harmonics 19 5.1. Specializations 20 arXiv:0809.2017v1 [math.OC] 11 Sep 2008 5.2. Linear programming bound for the unit sphere 21 5.3. Harmonic polynomials 22 5.4. Addition formula 25 Appendix A. Symmetric tensors and polynomials 26 Appendix B. Further reading 27 References 28 Date: September 11, 2008. 1991 Mathematics Subject Classification. 43A35, 52C17, 90C22, 94B65. Key words and phrases. semidefinite programming, harmonic analysis, symmetry reduction, block diagonalization, theta function, positive kernels, Peter-Weyl theorem, Bochner theorem, boolean harmonics, spherical harmonics. The author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4. 1 2 FRANK VALLENTIN 1. Introduction Semidefinite programming is a vast extension of linear programming and has a wide range of applications: combinatorial optimization and control theory are the most famous ones.
    [Show full text]
  • Norms of Zonal Spherical Functions and Fourier Series on Compact Symmetric Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOUHNAI. OF FUNCTIONAL ANALYSIS 44, 74-86 (1981) Norms of Zonal Spherical Functions and Fourier Series on Compact Symmetric Spaces B. DRESELER Fachbereich Mathematik, Universittit Siegen, 5900 Siegen, Federal Republic of Germany Communicated bjj the Editors New estimates are given for the p-norms of zonal spherical functions on compact symmetric spaces. These estimates are applied to give a Cohen type inequality for convoluter norms which leads to negative results on p-mean convergence of Fourier series on compact symmetric spaces of arbitrary rank. 1. INTRODUCTION In a recent paper 151 Dooley established estimates from below for p-norms of irreducible characters on compact Lie groups. Nice applications of these estimates to divergence results for Fourier series on compact Lie groups were proved by Giulini ef al. [8] (cf. also [9]). In [6, 71 a Cohen type inequality for Jacobi polynomials is proved and leads to the solution of the divergence problem for Jacobi expansions and zonal Fourier expansions on compact symmetric spaces of rank one. In this paper we prove the extension of some of the main results in the papers above to compact symmetric spaces of arbitrary rank. Also, in the special case of compact Lie groups our proofs are new and more in the spirit of paper [7]. Before stating our results explicitly we introduce the notations that will be used throughout the paper. General references for the harmonic analysis on compact symmetric spaces are [ 10, 11, 191.
    [Show full text]
  • Arxiv:1706.09047V1 [Math.RT] 21 Jun 2017 Iei.E-Mail: Nigeria
    On harmonic analysis of spherical convolutions on semisimple Lie groups. Olufemi O. Oyadare Abstract This paper contains a non-trivial generalization of the Harish- Chandra transforms on a connected semisimple Lie group G, with finite center, into what we term spherical convolutions. Among other results we show that its integral over the collection of bounded spherical functions at the identity element e ∈ G is a weighted Fourier transforms of the Abel transform at 0. Being a function on G, the restriction of this integral of its spherical Fourier trans- forms to the positive-definite spherical functions is then shown to be (the non-zero constant multiple of) a positive-definite dis- tribution on G, which is tempered and invariant on G = SL(2, R). These results suggest the consideration of a calculus on the Schwartz algebras of spherical functions. The Plancherel measure of the spherical convolutions is also explicitly computed. Subject Classification: 43A85, 22E30, 22E46 Keywords: Spherical Bochner theorem: Tempered invariant distributions: Harish-Chandra’s Schwartz algebras 1 Introduction Let G be a connected semisimple Lie group with finite center, and de- arXiv:1706.09047v1 [math.RT] 21 Jun 2017 note the Harish-Chandra-type Schwartz spaces of functions on G by Cp(G), 0 <p ≤ 2. We know that Cp(G) ⊂ Lp(G) for every such p, and if K is a max- imal compact subgroup of G such that Cp(G//K) represents the subspace of Cp(G) consisting of the K−bi-invariant functions, Trombi and Varadarajan [11.] have shown that the spherical Fourier transform f 7→ f is a linear topo- logical isomorphism of Cp(G//K) onto the spaces Z¯(Fǫ), ǫ = (2/p) − 1, b Department of Mathematics, Obafemi Awolowo University, Ile-Ife, 220005, Nigeria.
    [Show full text]
  • Orthogonal Polynomials Arising from the Wreath Products
    ORTHOGONAL POLYNOMIALS ARISING FROM THE WREATH PRODUCTS HIROKI AKAZAWA AND HIROSHI MIZUKAWA Abstract. Zonal spherical functions of the Gelfand pair (W (Bn), Sn) are expressed in terms of the Krawtchouk polynomials which are a special family of Gauss’ hypergeometric functions. Its gener- alizations are considered in this abstract. Some class of orthogonal polynomials are discussed in this abstract which are expressed in terms of (n + 1, m + 1)-hypergeometric functions. The orthogonal- ity comes from that of zonal spherical functions of certain Gelfand pairs of wreath product. Resum´ e´. Des fonctions sph´eriqueszonales de la paire (W (Bn),Sn) de Gelfand sont exprim´eesen termes de polynˆomexde Krawtchouk qui sont une famille sp´ecialedes fonctions hyperg´eom´etriquesdes Gauss. Ses g´en´eralisationssont consid´er´eesdans cet abstrait. Une certaine classe des polynˆomexorthogonaux sont discut´eesdans cet abstrait qui sont exprim´esen termes de fonctions (n + 1, m + 1)- hyperg´eom´etriques. L’orthogonalit´evient de celle des fonctions sph´eriqueszonales de certaines paires de Gelfand de produits en couronne. 1. Introduction Askey-Wilson polynomials and q-Racah polynomials are fundamen- tal orthogonal polynomials which are described by the basic hyper- geometric functions. Roughly speaking there are two points of view of orthogonal polynomials. One is through the Riemannian symmet- ric spaces which are homogeneous spaces of Lie groups. The other is through the finite groups. In this abstract we discuss some discrete orthogonal polynomials arising from Gelfand pairs [15, 16] of wreath products. A pair of groups (G, H) is called a Gelfand pair if the induced repre- G sentation 1H ∼= C(G/H), where C(G/H) is a complex valued functions defined over G/H, is multiplicity free as a G-module.
    [Show full text]
  • The Unramified Principal Series of P-Adic Groups. II. the Whittaker
    COMPOSITIO MATHEMATICA W. CASSELMAN J. SHALIKA The unramified principal series of p-adic groups. II. The Whittaker function Compositio Mathematica, tome 41, no 2 (1980), p. 207-231. <http://www.numdam.org/item?id=CM_1980__41_2_207_0> © Foundation Compositio Mathematica, 1980, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 41, Fasc. 2, 1980, pag. 207-231 © 1980 Sijthoff &#x26; Noordhoff International Publishers - Alphen aan den Rijn Printed in the Netherlands THE UNRAMIFIED PRINCIPAL SERIES OF p-ADIC GROUPS II THE WHITTAKER FUNCTION W. Casselman and J. Shalika Let G be a connected reductive algebraic group defined over the non-archimedean local field k. We will prove in this paper an explicit formula for a certain so-called Whittaker function associated to the unramified principal series of G(k), under the assumption that the group G is itself unramified - that is to say, arises by base extension to k from a smooth reductive group over the integers Q of k. This formula has been discovered independently by Shintani [8] when G = GLn and Kato [9] for Chevalley groups, and was also in fact conjectured by Langlands several years ago (in correspondence with Godement).
    [Show full text]
  • 13. the Spectral Side of the Selberg Trace Formula 39 14
    NOTES ON THE TRACE FORMULA RAHUL KRISHNA Abstract. These are notes for MATH 482-2, taught in Spring 2018 at Northwestern university. They are very rough, so please let me know of any comments or corrections at [email protected]. Contents 1. Number theoretic motivation 1 2. Some degenerate versions of the trace formula 5 3. Introduction to symmetric spaces 8 4. Harmonic analysis on symmetric spaces 10 5. Explicit spherical harmonics on symmetric spaces 13 6. Kernels and the trace formula for compact quotients 17 7. The trace formula for compact ΓnH 21 8. Non-compact quotients of H 24 9. Introduction to Eisenstein series 27 10. The spectral expansion: cuspidal part 31 11. Analytic continuation of Eisenstein series 34 12. Residues of Eisenstein series and the spectral expansion 37 13. The spectral side of the Selberg trace formula 39 14. The parabolic contribution and a first application 42 15. Weyl’s law and the existence of cusp forms 45 16. An application of the theory of Eisenstein series 48 17. The prime geodesic theorem 51 18. Adelic theory: motivations 54 19. Automorphic representations 57 20. Langlands’ reciprocity conjecture 60 21. Langlands’ functoriality conjecture 63 22. The simplest case of functoriality 65 References 66 1. Number theoretic motivation This first lecture will be mainly for motivation, so it may be a little all over the place. We will start from basics next time. 1.1. Why should we care about the trace formula? A fundamental (the fundamental?) problem in algebraic number theory is to find a good description of the absolute Galois group Gal(Q¯ =Q).
    [Show full text]