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2016 Matrix valued orthogonal polynomials for Gelfand pairs of rank one
Gert Heckman, Maarten van Pruijssen
Tohoku Math. J. (2) 68(3): 407-437 (2016). DOI: 10.2748/tmj/1474652266


In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair $(G,K)$ of rank one, as a generalization of earlier work by Koornwinder [30] and subsequently by Koelink, van Pruijssen and Roman [28], [29] for the pair (SU(2)$\times$SU(2), SU(2)), and by Grünbaum, Pacharoni and Tirao [13] for the pair (SU(3), U(2)). Our method is based on representation theory using an explicit determination of the relevant branching rules. Our matrix valued orthogonal polynomials have the Sturm-Liouville property of being eigenfunctions of a second order matrix valued linear differential operator coming from the Casimir operator, and in fact are eigenfunctions of a commutative algebra of matrix valued linear differential operators coming from the $K$-invariant elements in the universal enveloping algebra of the Lie algebra of $G$.


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Gert Heckman. Maarten van Pruijssen. "Matrix valued orthogonal polynomials for Gelfand pairs of rank one." Tohoku Math. J. (2) 68 (3) 407 - 437, 2016.


Received: 17 March 2014; Revised: 18 November 2014; Published: 2016
First available in Project Euclid: 23 September 2016

zbMATH: 1360.22019
MathSciNet: MR3550926
Digital Object Identifier: 10.2748/tmj/1474652266

Primary: 22E46
Secondary: 33C47

Keywords: matrix valued orthogonal polynomials , multiplicity free branching , Spherical varieties of rank one

Rights: Copyright © 2016 Tohoku University


Vol.68 • No. 3 • 2016
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