Harmonic analysis in the complex hyperbolic plane and the matrix valued hypergeometric function

P. ROMÁN; J. TIRAO

Leganés

Congreso; 11th International symposium on orthogonal polynomials, special functions and applications; 2011

For a locally compact group G with a compact subgroup K, the spherical transform on the convolution algebra C_{c,delta}(G) and its corresponding inversion formula are discussed. Here C_{c,delta}(G) denotes the the algebra of all continuous functions f with compact support on G such that \bar \chi_delta * f = f * \bar \chi_delta = f and \chi_delta denotes the character of a unitary irreducible representation of K times its dimension. The case of the pair G = SU(2;1) and K = U(2) is considered in full generality, i.e. for any K-type. Since we work with spherical functions of an arbitrary K-type, the spherical functions are now described explicitly in terms of matrix valued hypergeometric functions. In this way we obtain expressions for the spherical transform and the corresponding inversion formula which involve matrix hypergeometric functions of arbitrary size. We obtain an inversion formula for the spherical transform by using the Fourier inversion formula in G. If we consider K-types of dimension one the spherical transform reduces to a multiple of the Jacobi transform.