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Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
Maarten Van Pruijssena,
Pablo Románb a Universität Paderborn, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany
b CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria, Córdoba, Argentina
Abstract:
We present a method to obtain infinitely many examples of pairs
$(W,D)$ consisting of a matrix weight
$W$ in one variable and a symmetric second-order differential operator
$D$. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs
$(G,K)$ of rank one and a suitable irreducible
$K$-representation. The heart of the construction is the existence of a suitable base change
$\Psi_{0}$. We analyze the base change and derive several properties. The most important one is that
$\Psi_{0}$ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group
$G$ as soon as we have an explicit expression for
$\Psi_{0}$. The weight
$W$ is also determined by
$\Psi_{0}$. We provide an algorithm to calculate
$\Psi_{0}$ explicitly. For the pair $(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2))$ we have implemented the algorithm in GAP so that individual pairs
$(W,D)$ can be calculated explicitly. Finally we classify the Gelfand pairs
$(G,K)$ and the
$K$-representations that yield pairs
$(W,D)$ of size
$2\times2$ and we provide explicit expressions for most of these cases.
Keywords:
matrix valued classical pairs; multiplicity free branching.
MSC: 22E46;
33C47 Received: April 30, 2014; in final form
December 12, 2014; Published online
December 20, 2014
Language: English
DOI:
10.3842/SIGMA.2014.113
Fulltext:
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