A multivariate version of the disk convolution

TL;DR

This paper derives explicit product formulas for spherical functions related to certain Gelfand pairs and Hermitian symmetric spaces, extending known results from rank one to higher rank cases.

Contribution

It provides new explicit product formulas and hypergroup structures for multivariate spherical functions associated with Hermitian symmetric spaces, generalizing classical disk convolution results.

Findings

Explicit product formulas for spherical functions of Gelfand pairs.

Hypergroup structures parametrized by real p.

Extension of rank-one results to higher rank cases.

Abstract

We present an explicit product formula for the spherical functions of the compact Gelfand pairs $(G,K_1)= (SU(p+q), SU(p)\times SU(q))$ with $p\ge 2q$, which can be considered as the elementary spherical functions of one-dimensional $K$-type for the Hermitian symmetric spaces $G/K$ with $K= S(U(p)\times U(q))$. Due to results of Heckman, they can be expressed in terms of Heckman-Opdam Jacobi polynomials of type $BC_q$ with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real $p\in]2q-1,\infty[$. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman-Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for $q=1$ to higher rank.