The Spherical Transform of any k-Type in a Locally Compact Group
Journal of Lie theory, Tome 22 (2012) no. 2, pp. 361-395
Cet article a éte moissonné depuis la source Heldermann Verlag
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 ${}_2H_1$.
Classification :
33C45, 22E46
Mots-clés : Spherical transform, spherical functions, matrix hypergeometric function
Mots-clés : Spherical transform, spherical functions, matrix hypergeometric function
@article{JLT_2012_22_2_JLT_2012_22_2_a1,
author = {P. M. Rom\'an and J. Tirao},
title = {The {Spherical} {Transform} of any {k-Type} in a {Locally} {Compact} {Group}},
journal = {Journal of Lie theory},
pages = {361--395},
year = {2012},
volume = {22},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JLT_2012_22_2_JLT_2012_22_2_a1/}
}
P. M. Román; J. Tirao. The Spherical Transform of any k-Type in a Locally Compact Group. Journal of Lie theory, Tome 22 (2012) no. 2, pp. 361-395. http://geodesic.mathdoc.fr/item/JLT_2012_22_2_JLT_2012_22_2_a1/