Journal of Lie Theory

no. 2
The Spherical Transform of any k-Type in a Locally Compact Group
Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 361-395
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 ${}_2H_1$.
DOI: 10.5802/jolt.672
Classification: 33C45, 22E46
Keywords: Spherical transform, spherical functions, matrix hypergeometric function 
@article{JOLT_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},
     doi = {10.5802/jolt.672},
     zbl = {1255.33005},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.672/}
}
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P. M. Román; J. Tirao. The Spherical Transform of any k-Type in a Locally Compact Group. Journal of Lie Theory, Volume 22 (2012) no. 2, pp. 361-395. doi: 10.5802/jolt.672

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