Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Abstract

Let $p,q$ positive integers. The groups $U_p(\b C)$ and $U_p(\b C)\times U_q(\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\b C)\times \b R$ canonically as groups of automorphisms where $M_{p,q}(\b C)$ is the vector space of all complex $p\times q$-matrices. The associated orbit spaces may be identified with $\Pi_q\times \b R$ and $\Xi_q\times \b R$ respectively with the cone $\Pi_q$ of positive semidefinite matrices and the Weyl chamber $\Xi_q={x\in\b R^q: x_1\ge...\ge x_q\ge 0}$. In this paper we compute the associated convolutions on $\Pi_q\times \b R$ and $\Xi_q\times \b R$ explicitly depending on $p$. Moreover, we extend these convolutions by analytic continuation to series of convolution structures for arbitrary parameters $p\ge 2q-1$. This leads for $q\ge 2$ to continuous series of noncommutative hypergroups on $\Pi_q\times \b R$ and commutative hypergroups on $\Xi_q\times \b R$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on $\Pi_q$ and $\Xi_q$. In particular, we give a non-positive product formula for these Laguerre functions on $\Xi_q$. The paper extends the known case $q=1$ due to Koornwinder, Trimeche, and others as well as the group case with integers $p$ due to Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, it is closely related to product formulas for multivariate Bessel and other hypergeometric functions of R\"osler.