Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Volume 123 / 2011

Michael Voit Colloquium Mathematicum 123 (2011), 149-179 MSC: Primary 43A62; Secondary 33C67, 33C52, 43A90, 43A20. DOI: 10.4064/cm123-2-1


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.


  • Michael VoitFakultät Mathematik
    Technische Universität Dortmund
    Vogelpothsweg 87
    D-44221 Dortmund, Germany

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