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

P. ROMÁN; JUAN TIRAO

JOURNAL OF LIE THEORY

HELDERMANN VERLAG

Año: 2012 vol. 22 p. 361 - 395

0949-5932

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. The case of the group G=SU(2,1) and the compact subgroup K=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 2H1.