Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs

TL;DR

This paper introduces a method to deform matrix-valued orthogonal polynomials linked to Gelfand pairs, preserving their Sturm-Liouville property, and provides explicit formulas and new deformations for specific group-related polynomials.

Contribution

The paper develops a novel deformation technique for matrix-valued orthogonal polynomials associated with Gelfand pairs, including explicit formulas and applications to classical groups.

Findings

Explicit formula for the spherical function $$ in terms of Krawtchouk polynomials.

Deformation of $ ext{SU}(n)$ related polynomial families.

New family of polynomials with an extra free parameter for $ ext{Sp}(n)$.

Abstract

In this paper we present a method to obtain deformations of families of matrix-valued orthogonal polynomials that are associated to the representation theory of compact Gelfand pairs. These polynomials have the Sturm-Liouville property in the sense that they are simultaneous eigenfunctions of a symmetric second order differential operator and we deform this operator accordingly so that the deformed families also have the Sturm-Liouville property. Our strategy is to deform the system of spherical functions that is related to the matrix-valued orthogonal polynomials and then check that the polynomial structure is respected by the deformation. Crucial in these considerations is the full spherical function $\Psi_{0}$, which relates the spherical functions to the polynomials. We prove an explicit formula for $\Psi_{0}$ in terms of Krawtchouk polynomials for the Gelfand pair $(\mathrm{SU}(2)\times\mathrm{SU}(2),\mathrm{diag}(\mathrm{SU}(2)))$. For the matrix-valued orthogonal polynomials associated to this pair, a deformation was already available by different methods and we show that our method gives same results using explicit knowledge of $\Psi_{0}$. Furthermore we apply our method to some of the examples of size $2\times2$ for more general Gelfand pairs. We prove that the families related to the groups $\mathrm{SU}(n)$ are deformations of one another. On the other hand, the families associated to the symplectic groups $\mathrm{Sp}(n)$ give rise to a new family with an extra free parameter.