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Matrix Valued Orthogonal Polynomials Related to Compact Gel'fand Pairs of Rank

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Matrix Valued Orthogonal Polynomials Related to Compact Gel'fand Pairs of Rank Matrix valued orthogonal polynomials related to compact Gel'fand pairs of rank one ISBN 978-94-6191-511-5 NUR 921 c 2012 by M.L.A. van Pruijssen. All rights reserved. Matrix valued orthogonal polynomials related to compact Gel'fand pairs of rank one Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. mr. S.C.J.J. Kortmann, volgens besluit van het college van decanen in het openbaar te verdedigen op woensdag 19 december 2012 om 10.30 uur precies door Maarten Leonardus Antonius van Pruijssen geboren op 4 juni 1979 te Oosterhout Promotoren: prof. dr. G.J. Heckman prof. dr. H.T. Koelink Copromotor: dr. P. Rom´an Universidad Nacional de C´ordoba Manuscriptcommissie: prof. dr. M. Brion Universit´ede Grenoble 1 prof. dr. F.A. Gr¨unbaum University of California, Berkeley prof. dr. T.H. Koornwinder Universiteit van Amsterdam prof. dr. N.P. Landsman prof. dr. P.-E. Paradan Universit´eMontpellier 2 Contents 1 Introduction 1 1.1 Orthogonal polynomials . .1 1.2 Realization as matrix coefficients . .4 1.3 The spectral problem . .6 1.4 The general construction in a nutshell . 10 1.5 Applications and further studies . 11 1.6 This dissertation . 12 I The General Framework 15 2 Multiplicity Free Systems 17 2.1 Introduction . 17 2.2 Multiplicity free systems . 21 2.3 Inverting the branching rule . 31 2.4 Module structure . 51 3 Matrix Valued Polynomials 57 3.1 Introduction . 57 3.2 Multiplicity free triples . 58 3.3 Spherical functions . 60 3.3.1 Spherical functions and representations . 60 3.3.2 Recurrence relations for the spherical functions . 63 3.3.3 The space of spherical functions . 65 3.4 Spherical functions restricted to A ...................... 67 3.4.1 Transformation behavior . 67 3.4.2 Orthogonality and recurrence relations . 68 3.4.3 Differential operators . 71 3.5 Spherical polynomials . 74 3.5.1 Spherical polynomials on G ...................... 74 3.5.2 Spherical polynomials restricted to A ................. 76 v Contents 3.6 Full spherical polynomials . 78 3.6.1 Construction . 78 3.6.2 Properties . 78 3.6.3 Comparison to other constructions . 80 II The Example (SU(2) × SU(2); diag) 83 4 MVOPs related to (SU(2) × SU(2); diag), I 85 4.1 Introduction . 85 4.2 Spherical Functions of the pair (SU(2) × SU(2); diag) . 87 4.3 Recurrence Relation for the Spherical Functions . 90 4.4 Restricted Spherical Functions . 94 4.5 The Weight Matrix . 97 4.6 MVOPs associated to (SU(2) × SU(2); diag) . 102 4.6.1 Matrix valued orthogonal polynomials . 102 4.6.2 Polynomials associated to SU(2) × SU(2) . 104 4.6.3 Symmetries of the weight and the matrix polynomials . 104 4.7 Matrix Valued Differential Operators . 107 4.7.1 Symmetric differential operators . 107 4.7.2 Matrix valued differential operators for the polynomials Pn .... 108 4.8 Examples . 112 4.8.1 The case ` = 0; the scalar weight . 112 1 4.8.2 The case ` = 2 ; weight of dimension 2 . 113 4.8.3 Case ` = 1; weight of dimension 3 . 114 4.8.4 Case ` = 3=2; weight of dimension 4 . 118 4.8.5 Case ` = 2; weight of dimension 5 . 120 4.A Transformation formulas . 122 4.B Proof of the symmetry for differential operators . 126 5 MVOPs related to (SU(2) × SU(2); diag), II 131 5.1 Introduction . 131 5.2 LDU-decomposition of the weight . 135 5.3 MVOP as eigenfunctions of DO . 139 5.4 MVOP as MVHGF . 141 5.5 Three-term recurrence relation . 145 5.6 MVOP related to Gegenbauer and Racah . 149 5.7 Group theoretic interpretation . 156 5.7.1 Group theoretic setting of the MVOPs . 156 5.7.2 Calculation of the Casimir operators . 159 5.A Proof of Theorem 5.2.1 . 165 5.B Moments . 168 vi Contents Bibliography 169 Index 177 Samenvatting 179 Dankwoord 181 Curriculum Vitae 183 vii Chapter 1 Introduction In this dissertation we present a construction of matrix valued polynomials in one vari- able, with special properties, out of matrix coefficients on certain compact groups. The construction generalizes the theory that relates Jacobi polynomials to certain matrix co- efficients on compact groups. In the present chapter we motivate the research in this dissertation and we give an outline of the results. To this end, we discuss in Sections 1.1 and 1.2 the notion of matrix valued orthogonal polynomials and how their simplest examples, the scalar valued orthogonal polynomials, are related to matrix coefficients on compact Lie groups. Once the definitions and notations are fixed we can formulate the research goal. In Section 1.3 we discuss a spectral problem in the theory of Lie groups. In Section 1.4 we indicate that for the solutions of the spectral problem there is a general framework of certain functions, whose structure is determined by a family of matrix valued orthogonal polynomials. In Section 1.5 we discuss the use of this construction and some of the open questions. We close this chapter with an overview of the contents of the various chapters in Section 1.6. 1.1 Orthogonal polynomials 1.1.1. Let I = (a; b) ⊂ R be an interval, possibly unbounded. Let w : I ! R be a non- R negative integrable function satisfying I w(x)dx > 0, where dx is the Lebesgue measure R n and suppose that the moments are finite, i.e. I jx jw(x)dx < 1 for all n 2 N. Define the inner product h·; ·iw on the space of complex valued continuous functions on I by Z hf; hiw = f(x)h(x)w(x)dx: I A sequence of orthogonal polynomials on I with respect to h·; ·iw is a sequence fpd : d 2 Ng with pd(x) 2 C[x] of degree d satisfying hpd; pd0 iw = cdδd;d0 , with cd 2 R a positive number. A family of orthogonal polynomials satisfies a three term recurrence relation, i.e. there 1 Chapter 1. Introduction are sequences of complex numbers fad : d 2 Ng; fbd : d 2 Ng and fcd : d 2 Ng with the ad 6= 0 for which the functional equation xpd(x) = adpd+1(x) + bdpd(x) + cdpd−1(x) (1.1) holds on I. Conversely, given a three term recurrence relation (1.1) one can construct a sequence of polynomials fpd(x): d 2 Ng. Favard's theorem gives sufficient conditions on the coefficients ad; bd and cd to guarantee the existence of a positive Borel measure µ so that fpd(x): d 2 Ng is a sequence of orthogonal polynomials with respect to µ. See e.g. [Chi78] for an introduction to the theory of orthogonal polynomials. The classical orthogonal polynomials of Hermite, Laguerre and Jacobi are also eigen- functions of a second order differential operator that is symmetric with respect to h·; ·iw and Bochner showed that this additional property characterizes them among all the fam- ilies of orthogonal polynomials. 1.1.2. Among the generalizations of the theory of orthogonal polynomials is the theory of matrix valued orthogonal polynomials in one variable. Matrix valued polynomials are elements in C[x] ⊗ End(V ) where V is a finite dimensional complex vector space. The space C[x] ⊗ End(V ) is a bimodule over the matrix algebra End(V ). The Hermitian adjoint of A 2 End(V ) is denoted by A∗. Let I = (a; b) ⊂ R be an interval, possibly unbounded. A matrix weight W on I is an End(V )-valued function for which all its matrix entries are integrable functions such that W (x) is positive definite almost everywhere on I. Suppose that W has finite moments, R n i.e., that I jxj jWi;j(x)jdx < 1 for all i; j and all n 2 N, where the integration is entry wise. Define the pairing h·; ·iW : C[x] ⊗ End(V ) × C[x] ⊗ End(V ) ! End(V ) R ∗ by hP; QiW = I P (x) W (x)Q(x)dx. The pairing h·; ·iW is called an End(V )-valued inner product, i.e. it has the following properties. •h P; QS + RT iW = hP; QiW S + hP; RiW T for all S; T 2 End(V ) and all P; Q; R 2 C[x] ⊗ End(V ), ∗ •h P; QiW = hQ; P iW for all P; Q 2 C[x] ⊗ End(V ), •h P; P iW ≥ 0 for all P 2 C[x] ⊗ End(V ), i.e. hP; P iW is positive semi-definite. If hP; P iW = 0 then P = 0. The right End(V )-module C[x]⊗End(V ) with the pairing h·; ·iW is called a pre-Hilbert module, see e.g. [Lan95] and [RW98]. A family fPd : d 2 Ng with Pd 2 C[x] ⊗ End(V ) is called a family of matrix valued orthogonal polynomials for h·; ·iW if (i) deg Pd = d, (ii) the leading coefficient of Pd is non-singular and (iii) hPd;Pd0 iW = Sdδd;d0 with Sd 2 End(V ). Note that Sd is positive definite. The existence of a family of matrix valued orthogonal polynomials for a given matrix weight W (x) is proved in e.g. [GT07, Prop. 2.4], by a generalization of the Gram-Schmidt orthogonalization. 2 1.1. Orthogonal polynomials An important property of a family of matrix valued orthogonal polynomials is that they satisfy a three term recurrence relation, i.e. there are sequences fAd : d 2 Ng; fBd : d 2 Ng and fCd : d 2 Ng in End(V ) with the Ad invertible, such that xPd(x) = Pd+1(x)Ad + Pd(x)Bd + Pd−1(x)Cd holds on I.
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