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Let p,q be positive integers. The groups $U_{p} (ℂ)$ and $U_{p} (ℂ) \times U_{q} (ℂ)$ act on the Heisenberg group $H_{p, q} : = M_{p, q} (ℂ) \times ℝ$ 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 $Π_{q} \times ℝ$ and $Ξ_{q} \times ℝ$ respectively, $Π_{q}$ being the cone of positive semidefinite matrices and $Ξ_{q}$ the Weyl chamber $x \in ℝ^{q} : x ₁ \geq \dots \geq x_{q} \geq 0$ . In this paper we compute the associated convolutions on $Π_{q} \times ℝ$ and $Ξ_{q} \times ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution...

Induced representations and duality results for commutative hypergroups.

Michael Voit; Peter Hermann — 1995

Forum mathematicum

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A limit relation for Dunkl-Bessel functions of type A and B.

Compact Groups Having Almost Discrete Orbit Hypergroups.

Positive Characters on Commutative Hypergroups and Some applications.

Duals of subhypergroups and quotients of commutative hypergroups.

Strong Laws of Large Numbers for Random Walks Associated with a Class of One-Dimensional Convolution Structures.

A Law of the Iterated Logarithm for a Class of Polynomial Hypergroups.

On the Fourier Transformation of Positive, Positive Definite Measure on Commutative Hypergroups, and Dual Convolution Structures.

A positivity result and normalization of positive convolution structures.

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Induced representations and duality results for commutative hypergroups.