Spherical functions associated to the three dimensional hyperbolic space

F. A. GRUNBAUM, I. PACHARONI, J.TIRAO

INTERNATIONAL JOURNAL OF MATHEMATICS

WORLD SCIENTIFIC PUBL CO PTE LTD

Año: 2002 vol. 13 p. 727 - 784

0129-167X

The main purpose of this paper is to compute all irreducible spherical functions on G = SL(2; C) of arbitrary type 2 ^K , where K = SU(2). This is accomplished by associating to a spherical function on G a matrix valued function H on the three dimensional hyperbolic space H = G=K. The entries of H are solutions of two coupled systems of ordinary dierential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of H in terms of Gauss´ functions 2F1. Just as in the compact instance treated in [7], there is a useful role for a special class of generalized hypergeometric functions p+1Fp.