Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Michael Voit. "Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials." Colloquium Mathematicae 123.2 (2011): 149-179. <http://eudml.org/doc/284095>.

@article{MichaelVoit2011,
abstract = {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.},
author = {Michael Voit},
journal = {Colloquium Mathematicae},
keywords = {Heisenberg convolution; matrix cones; Weyl chambers; multivariate Laguerre polynomials; multivariate Bessel functions; product formulas; hypergroups; hypergroup characters},
language = {eng},
number = {2},
pages = {149-179},
title = {Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials},
url = {http://eudml.org/doc/284095},
volume = {123},
year = {2011},
}

TY - JOUR
AU - Michael Voit
TI - Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials
JO - Colloquium Mathematicae
PY - 2011
VL - 123
IS - 2
SP - 149
EP - 179
AB - 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.
LA - eng
KW - Heisenberg convolution; matrix cones; Weyl chambers; multivariate Laguerre polynomials; multivariate Bessel functions; product formulas; hypergroups; hypergroup characters
UR - http://eudml.org/doc/284095
ER -