Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials
Tom 123 / 2011
Streszczenie
Let $p,q$ be positive integers. The groups $U_p(\mathbb{C})$ and $U_p(\mathbb{C})\times U_q(\mathbb{C}) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\mathbb{C})\times \mathbb{R}$ canonically as groups of automorphisms, where $M_{p,q}(\mathbb{C})$ is the vector space of all complex $p\times q$ matrices. The associated orbit spaces may be identified with ${\mit\Pi}_q\times \mathbb{R}$ and ${\mit\Xi}_q\times \mathbb{R}$ respectively, ${\mit\Pi}_q$ being the cone of positive semidefinite matrices and ${\mit\Xi}_q$ the Weyl chamber $\{x\in\mathbb{R}^q : x_1 \ge\cdots\ge x_q\ge 0\}$. In this paper we compute the associated convolutions on ${\mit\Pi}_q\times \mathbb{R}$ and ${\mit\Xi}_q\times \mathbb{R}$ 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 ${\mit\Pi}_q\times \mathbb{R}$ and commutative hypergroups on ${\mit\Xi}_q\times \mathbb{R}$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on ${\mit\Pi}_q$ and ${\mit\Xi}_q$. In particular, we give a nonpositive product formula for these Laguerre functions on ${\mit\Xi}_q$. The paper extends the known case $q=1$ due to Koornwinder, Trimèche, and others, as well as the group case with integers $p$ due to Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, our results are closely related to product formulas for multivariate Bessel and other hypergeometric functions of Rösler.
