Zonal Spherical Functions on Some Symmetric Spaces

Abstract

Let G be a real semisimple Lie group with finite center, and K a maximal compact subgroup of G. A zonal spherical function on the symmetric space X=G/K is an simulatneous eigeiifunction (p(x) of all the invariant differential operators on X satisfying (p(kx) = cp(x) for any x^X, k^K, and (p(eK) =1, where e is the identity element in G. By the Cartan decomposition G = KAK, (p(x) is considered as a function on A. And by the separation of variables, we obtain differential operators on A from the invariant differential operators, which are called their radial components. In this paper, we investigate the radial components of the invariant differential operators and the zonal spherical functions when G is a real, complex or quanternion unimodular group. The eigenvalues of the zonal spherical functions is parametrized by the element in a*. Therefore, the system of differential equations on A satisfied by the zonal spherical function has as many parameters as dim a. However, we can construct a new system of differential equations which admits the other parameter V. It is shown that the zonal spherical function on the real, complex or quaternion unimodular group corresponds to the case in which V = -o-, 1, 2, respectively.