DOCSLIB.ORG

Finite Dimensional Representations of the Quantum Group

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Dimensional Representations of the Quantum Group Japan. J. Math. Vol. 19, No. 1, 1993 Finite dimensional representations of the quantum group GLq(n;C) and the zonal spherical functions on Uq(n-1)•_Uq(n) By Masatoshi NOUMI, Hirofumi YAMADA and Katsuhisa MIMACHI (Received March 16, 1992) •˜ 0. Introduction Representation theory of quantum groups is now studied in connection with various fields of mathematics. In this paper, we discuss representations of quantum groups and their homogeneous spaces from the viewpoint of spherical functions. In this direction, the spherical functions on the quantum group SUq(2) are studied by Vaksman-Soibelman [VS1], Koornwinder [K1, 2], Masuda et al. [MM1, 2] and Noumi-Mimachi [NM1, 2]. This paper, a detailed version of the announcement [NYM], deals with the quantum general linear group GLq(n;C) and its "compact" real form Uq(n), the quantum unitary group; the objects of our study are their algebras of functions A(GLq(n;C)) and A(Uq(n)). The main points we will discuss are the following: (1) An analogue of the Borel-Weil construction of finite dimensional irre ducible representations of GLq(n;C). (2) Irreducible decomposition of A(GLq(n;C)) as a two-sided A(GLq(n;C)) - comodule and the orthogonal decomposition of A(Uq(n)). (3) Invariant functional and the zonal spherical functions on the quantum homogeneous space Uq(n-1)•_Uq(n). We will give here a brief description of the contents without precise definition of the terminology. (1) For each dominant integral weight A, we realize the finite dimensional irreducible representation of GLq(n;C) of highest weight A as a space of relative invariants in A(GLq(n;C)) with respect to the Borel subgroup Bn , and also its basis consisting of standard monomials (Theorem 2.4 and 2.5). This gives a quantum analogue of the space of sections of the associated line bundle on the flag manifold. (2) We give the irreducible decomposition of A(GLq(n;C)) as a two-sided A(GLq(n;C))-comodule (Theorem 2.11), by using the complete reducibility the orem of finite dimensional Uq(si(n))-modules, due to Rosso [R] and Lusztig [L]. 32 M. NOUMI, H. YAMADA and K. MIMACHI This leads to the existence of an invariant functional on A(GLq(n;C)) and the parametrization of all finite dimensional irreducible A(GLq(n;C))-comodules by means of the dominant integral weights (Theorem 2.12). The coordinate ring A(Uq(n)) is defined as a Hopf *-algebra (A(GLq(n;C)),*). We introduce two Her mitian forms on A(Uq(n)) by means of the invariant functional. The irreducible decomposition of A(Uq(n)) is orthogonal with respect to these Hermitian forms (Theorem 3.4). This statement on Uq(n) is an algebraic reformulation of results of Woronowicz [W]. (3) The quantum homogeneous space Uq(n-1)•_Uq(n) is a quantum (2n-1) - sphere. The algebra of functions on this quantum homogeneous space is defined as the *-subalgebra of all left Uq(n-1)-invariants in A(Uq(n)) . We also give its irre ducible decomposition as a right A(Uq(n))-comodule (Proposition 4.3). Furthermore we determine the zonal spherical functions on Uq(n-1)•_Uq(n) explicitly in terms of the little q-Jacobi polynomials (Theorem 4.7). It is remarkable that the algebra of functions on the quotient space Uq(n-1)•_Uq(n)/H by the diagonal subgroup H of Uq(n) falls into a commutative algebra. On this commutative subalgebra, the invariant functional on Uq(n-1)•_Uq(n) is expressed by an iterated Jackson integral (Theorem 4.6). After we completed this work, we received preprints by Taft-Towber [TT], Hashimoto-Hayashi [HH] and Vaksman-Soibelman [VS2]. The first two, [TT] and [HH], also deal with the standard monomial bases for finite dimensional irreducible representations of GLq(n;C). Our arguments are different from theirs in the sense that we give a characterization of relative B-n-invariants (Theorem 2.2). The last one [VS2] is also devoted to a harmonic analysis on the odd dimensional quan tum spheres Uq(n-1)•_Uq(n). In our formulation, the zonal spherical functions are realized as elements of the algebra of functions. Throughout this paper we always assume that the parameter q •¸ E C•_{0} is not a root of unity. When the real form Uq(n) is concerned, we assume that q is a real number with 0<|q|<1. Contents •˜ 1. Quantum group GLq(n;C). 1.1. Quantum semigroup Matq(n;C). 1.2. Quantum minor determinants. 1.3. Quantum group GLq(n;C). 1.4. Quantum universal enveloping algebra Uq(g.?(n;C)). 1.5. A monomial basis for A(Matq(m,n;C)). •˜ 2. Finite dimensional representations of GLq(n;C). 2.1. Quantum G-spaces and relative invariants. 2.2. Representations of GLq(n;C). Quantum group GLq(n;C) and zonal spherical functions 33 2.3. Irreducibility of the representations A(B 1 GLq (n; C); zA). 2.4. Irreducible decomposition of A(GLq(n;C)). •˜ 3. Quantum group Uq(n) and the invariant functional. 3.1. Quantum group Uq(n) and its unitary representations. 3.2. Invariant functional.•˜ 4. Zonal spherical functions on Uq(n-1)•_Uq(n). 4.1. Quantum homogeneous space Uq(n-1)•_Uq(n). 4.2. Invariant functional on A(Uq(n-1)•_Uq(n)). 4.3. Zonal spherical functions on Uq(n-1)•_Uq(n). •˜ 1. Quantum group GLq(n;C) 1.1. Quantum semigroup Matq(n;C) For any positive integers m and n, we denote by Mat(m,n,C) the space of all m•~n complex matrices and by Mat(n;C) the space of all n•~n square matrices. We introduce the quantum semigroups Matq (n;C) and their action on the quantum spaces Matq(m,n;C). Hereafter we fix a nonzero complex number q. By definition, the coordinate ring A(Matq(m,n;C)) is the C-algebra generated by mn letters x2j (1 < i < m, 1 < j < n) with fundamental relations (1.1.a) (1.1.b) (1.1.c) When m=n, we write simply Matq(n;C)=Matq(n,n;C). One can directly verify that there exist unique C-algebra homomorphisms L : A(Matq(rn, n, C)) -~ A(Matq(m, r, C)) ® A(Matq(r, n, C)) and : A(Matq(n, C)) --~ C satisfying (1.2) It is clear that each A(Matq(n;C)) becomes a bialgebra with coproduct •¢ and counit ƒÃ, and that each A(Matq(m,n;C)) has the natural structure of two-sided (A(Matq(m;C)), A(Matq(n;C)))-comodule. We remark that the commutation relations (1.1) of the algebra A(Matq(n;C)) are related to the so-called the constant R-matrix of type An-1(see[RTF]). Define the matrix R •¸ Mat(n;C) ® Mat(n;C) by (1.3) 34 M. NOUMI,H. YAMADAand K. MIMACHI where Eij are the matrix units. It is well-knownthat this matrix R satisfies the Yang-Baxter equation (see[J1]). With the matrix notation the commutation relations (1.1) are simply expressed in the form (1.4) where X ® X = xzaxkQE2~ ® Ei As preliminaries to the representation theory of the quantum group GLq(n;C), we investigate some fundamental comodules over the bialgebra A(Matq(n;C)). In the following, we will sometimes write Mn for (Matq(n;C). Let S be the C-algebra generated by n letters x1, ..., xn with fundamental relations Then one can directly show that there exists a unique C-algebra homomorphism RMn: S --* S ® A(Matq(n, C)) such that (1.5) Then the algebra S=C [x1,..., xn] becomes a right A(Matq (n;C))-comodule with this homomorphism RM. Note that the algebra S has a graduation S = ~r>o Sr, where Sr denotes the subspace of S of all homogeneous elements of degree r. It is clear that the subspaces Sr(r•¸N) are right A(Matq(n;C))-subcomodules of S. We call the right comodule Sr the symmetric tensor representation of the quantum semigroup (Matq(n;C) of degree r. The comodule S1 = Cx1® ... Cxn is also called the vector representation. Let E be the C-algebra generated by n letters y1,...,yn with (anti-) commuta tion relations Then one can directly show that there exists a unique C-algebra homomorphism RMn : E --~E ® A(Matq(n; C)) satisfying (1.6) and that E becomes a right A(Matq(n;C))-comodule with this coaction. Similarly to the case of S, the right A(Matq(n;C))-comodule E decomposes into a direct Quantumgroup GLq(n;C) and zonalspherical functions 35 sum E = ® o Er of subcomodules according to the homogeneous degree. The right A(Matq(n;C))-comodule Er is called the alternating tensor representation of (Matq(n;C) of degree r; the comoduleE1 is isomorphic to the vector representation S1. Note that the subspace Er of degree r has a C-basis {Yji• • • yjr;1 <j1 <.<•• jr <n}: We will see later that the representations Sr(r•¸N) and Er (0 < r < n) are irreducible. REMARK. The "exterior algebra" E=C[y1,...,yn] also has a structure of left A(Matq(n;C))-comodule whose coaction is given by the C-algebra homomor phism LM : E -~ A(Matq(n; C)) ® E such that Then the subspaces Er (0 < r < n) are again left subcomodules of E. •¡ 1.2. Quantum minor determinants We now investigate the matrix elements of the right comodule Er (0 < r < n).
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]