Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Michael Voit¹

¹ Fakultät Mathematik Technische Universität Dortmund Vogelpothsweg 87 D-44221 Dortmund, Germany

Colloquium Mathematicum, Tome 123 (2011) no. 2, pp. 149-179

Résumé

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.

Détail

DOI : 10.4064/cm123-2-1

Keywords: positive integers groups mathbb mathbb times mathbb act heisenberg group mathbb times mathbb canonically groups automorphisms where mathbb vector space complex times matrices associated orbit spaces may identified mit times mathbb mit times mathbb respectively mit being cone positive semidefinite matrices mit weyl chamber mathbb cdots paper compute associated convolutions mit times mathbb mit times mathbb explicitly depending moreover extend these convolutions analytic continuation series convolution structures arbitrary parameters q leads continuous series noncommutative hypergroups mit times mathbb commutative hypergroups mit times mathbb latter describe dual space terms multivariate laguerre bessel functions mit mit particular nonpositive product formula these laguerre functions mit paper extends known due koornwinder trim che others group integers due faraut benson jenkins ratcliff others moreover results closely related product formulas multivariate bessel other hypergeometric functions sler