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HOME > Table of Contents and Abstracts > Vol. 68, No. 3
Tohoku Mathematical Journal
2016
September
SECOND SERIES VOL. 68, NO. 3
Tohoku Math. J.
68 (2016), 407-437
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Title
MATRIX VALUED ORTHOGONAL POLYNOMIALS FOR GELFAND PAIRS OF RANK ONE
Author
Gert Heckman and Maarten van Pruijssen
(Received March 17, 2014, revised November 18, 2014) |
Abstract.
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 $G$.
Mathematics Subject Classification.
Primary 22E46; Secondary 33C47.
Key words and phrases.
Spherical varieties of rank one, multiplicity free branching, matrix valued orthogonal polynomials.
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