Geodesic
The Spherical Transform of any k-Type in a Locally Compact Group
Journal of Lie theory, Tome 22 (2012) no. 2, pp. 361-395.

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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 
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     title = {The {Spherical} {Transform} of any {k-Type} in a {Locally} {Compact} {Group}},
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P. M. Rom�n; 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/