Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

TL;DR

This paper develops explicit convolution formulas on matrix cones related to Heisenberg groups, extending to hypergroup structures for arbitrary parameters, and connects these to multivariate Laguerre and Bessel functions.

Contribution

It introduces new convolution structures on matrix cones depending on a parameter p, extending classical cases and linking to multivariate special functions.

Findings

Explicit convolution formulas on $Pi_q imes R$ and $Xi_q imes R$ depending on p.

Extension of convolutions to series for all p ≥ 2q-1.

Identification of dual spaces via multivariate Laguerre and Bessel functions.

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.