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Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Michael Voit (2011)

Colloquium Mathematicae

Let p,q be positive integers. The groups U p ( ) and U p ( ) × U q ( ) act on the Heisenberg group H p , q : = M p , q ( ) × canonically as groups of automorphisms, where M p , q ( ) is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with Π q × and Ξ q × respectively, Π q being the cone of positive semidefinite matrices and Ξ q the Weyl chamber x q : x x q 0 . In this paper we compute the associated convolutions on Π q × and Ξ q × explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution...

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