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2011 | 123 | 2 | 149-179
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Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

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Let p,q be positive integers. The groups $U_{p}(ℂ)$ and $U_{p}(ℂ) × U_q(ℂ)$ act on the Heisenberg group $H_{p,q}: = M_{p,q}(ℂ) × ℝ$ canonically as groups of automorphisms, where $M_{p,q}(ℂ)$ is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with $Π_q × ℝ$ and $Ξ_q × ℝ$ respectively, $Π_q$ being the cone of positive semidefinite matrices and $Ξ_q$ the Weyl chamber ${x ∈ ℝ^q: x₁ ≥ ⋯ ≥ x_q ≥ 0}$. In this paper we compute the associated convolutions on $Π_q × ℝ$ and $Ξ_q × ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution structures for arbitrary parameters p ≥ 2q-1. This leads for q ≥ 2 to continuous series of noncommutative hypergroups on $Π_q × ℝ$ and commutative hypergroups on $Ξ_q × ℝ$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on $Π_q$ and $Ξ_q$. In particular, we give a nonpositive product formula for these Laguerre functions on $Ξ_q$. 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.
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  • Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-2-1
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