Matrix valued classical pairs related to compact Gelfand pairs of rank one

P. ROMÁN; M. VAN PRUIJSSEN

SYMMETRY, INTEGRABILITY AND GEOMETRY

NATL ACAD SCI UKRAINE

Año: 2014 vol. 10 p. 1 - 28

1815-0659

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 (USp(2n),USp(2n-2)xUSp(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 2x2 and we provide explicit expressions for most of these cases.