Separable representations for automorphism groups of infinite symmetric spaces

In this paper we consider the separable unitary representations for the automorphism groups of the classical infinite rank (Finsler) symmetric spaces defined by Schatten p-classes (often referred to as restricted groups). Following earlier work of Ol'shanskii and Voiculescu, it is shown that the spherical representations are always type I, the form of the irreducible spherical functions is determined, and their analyticity established. Using an intuitive geometric argument, it is shown that the real spherical functions extend to the Hilbert-Schmidt limit and never beyond. This yields a complete determination of the separable representations for groups corresponding to p-classes with p > 2.