Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Let $p,q$ positive integers. The groups $U_p(\text{b}C)$ and $U_p(\b C)\times U_q(\b C) $actontheHeisenberggroup$H_{p,q}:=M_{p,q}(\b C)\times \b R$canonicallyasgroupsofautomorphismswhere$M_{p,q}(\b C)$ is the vector space of all complex $p\times q$-matrices. The associated orbit spaces may be identified with ${\mathrm{\Pi }}_q\times \text{b}R$ and ${\mathrm{\Xi }}_q\times \text{b}R$ respectively with the cone ${\mathrm{\Pi }}_q$ of positive semidefinite matrices and the Weyl chamber ${\mathrm{\Xi }}_q=x\in \text{b}{R}^q:x_1\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge x_q\ge 0$. In this paper we compute the associated convolutions on ${\mathrm{\Pi }}_q\times \text{b}R$ and ${\mathrm{\Xi }}_q\times \text{b}R$ 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 ${\mathrm{\Pi }}_q\times \text{b}R$ and commutative hypergroups on ${\mathrm{\Xi }}_q\times \text{b}R$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on ${\mathrm{\Pi }}_q$ and ${\mathrm{\Xi }}_q$. In particular, we give a non-positive product formula for these Laguerre functions on ${\mathrm{\Xi }}_q$. 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.