Spherical functions, the complex hyperbolic plane and the hypergeometric operator

PABLO ROMÁN; JUAN TIRAO

INTERNATIONAL JOURNAL OF MATHEMATICS

World Scientific Publishing Company

Año: 2006 p. 1 - 23

0129-167X

In this paper we determine all irreducible spherical functions Phi of any K-type associated to the dual Hermitian symmetric pairs (G,K)=(SU(3),U(2)) and (SU(2,1),U(2)). This is accomplished by associating to Phi a vector valued function H=H(u) of a real variable u, analytic at u=0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair (SU(3),U(2)) this project was started in [4].