dc.contributor.author Koelink, Erik dc.contributor.author van Pruijssen, Maarten dc.contributor.author Román, Pablo Manuel dc.date.accessioned 2017-09-07T08:07:15Z dc.date.available 2017-09-07T08:07:15Z dc.date.issued 2017-06-14 dc.identifier.uri http://publications.mfo.de/handle/mfo/1304 dc.description Research in Pairs 2017 en_US dc.description.abstract In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary en_US rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case SU($n$+ 1) meets all the conditions that we impose in Part 1. For any $k \in \mathbb{N}_0$ we obtain families of orthogonal polynomials in n variables with values in the $N \times N$-matrices, where $N = \binom{n+k}{k}$ The case $k = 0$ leads to the classical Heckman-Opdam polynomials of type $A_n$ with geometric parameter. For $k = 1$ we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever $n ≥ 2$. We also give explicit expressions of the spherical functions that determine the matrix weight for $k = 1$. These expressions are used to calculate the spherical functions that determine the matrix weight for general $k$ up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case $n = 1$. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for $(n, k)$ equal to $(2, 1)$ and $(3, 1)$. dc.language.iso en_US en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2017,16 dc.title Matrix Elements of Irreducible Representations of SU(n+1) x SU(n+1) and Multivariable Matrix-Valued Orthogonal Polynomials en_US dc.type Preprint en_US dc.rights.license Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. de dc.rights.license This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. en dc.identifier.doi 10.14760/OWP-2017-16 local.scientificprogram Research in Pairs 2017 en_US local.series.id OWP-2017-16 dc.identifier.urn urn:nbn:de:101:1-20170807728 dc.identifier.ppn 165404959X
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