The Spherical Transform of any k-Type in a Locally Compact Group

Pablo Manuel Román^1, 2 ; Juan Tirao²

¹Dept. of Mathematics, Katholieke Universiteit, Celestijnenlaan 200b - bus 2400, 3001 Leuven, Belgium
²CIEM -- FaMAF, Universidad Nacional, Medina Allende s/n, Ciudad Universitaria, Córdoba, Argentina

Journal of Lie Theory, Tome 22 (2012) no. 2, pp. 361-395

Résumé

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 ${}_2H_1$.

Détail

Classification : 33C45, 22E46
Mots-clés : Spherical transform, spherical functions, matrix hypergeometric function