Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 22 (2012), No. 2, 361--395Copyright Heldermann Verlag 2012 The Spherical Transform of any k-Type in a Locally Compact Group Pablo Manuel Román Dept. of Mathematics, Katholieke Universiteit, Celestijnenlaan 200b - bus 2400, 3001 Leuven, Belgium and: CIEM -- FaMAF, Universidad Nacional, Medina Allende s/n, Ciudad Universitaria, Córdoba, Argentina roman@mate.uncor.edu Juan Tirao CIEM -- FaMAF, Universidad Nacional, Medina Allende s/n, Ciudad Universitaria, Córdoba, Argentina tirao@mate.uncor.edu [Abstract-pdf] Given a locally compact group $G$ and a compact subgroup $K$, we develop and study a spherical transform on the convolution algebra $C_{c,\delta}(G)$ of all continuous functions $f$ with compact support on $G$ such that $\overline \chi_\delta\ast f=f\ast \overline \chi_\delta=f$. Here $\chi_\delta$ denotes the character of a unitary irreducible representation of $K$ times its dimension. We obtain an inversion formula for the spherical transform by using the Fourier inversion formula in $G$. \hfill\break The case of the group $G={\rm SU}(2,1)$ and the compact subgroup $K={\rm U}(2)$ is discussed in detail. We give explicit expressions for the spherical transform and the corresponding inversion formula in terms of the matrix hypergeometric function ${}_2H_1$. Keywords: Spherical transform, spherical functions, matrix hypergeometric function. MSC: 33C45, 22E46 [ Fulltext-pdf  (434  KB)] for subscribers only.