Spherical functions and local densities on hermitian forms

Dedicated to Professor Ichiro Satake on his seventieth birthday

Abstract

First we give a formula of spherical functions on certain spherical homogeneous spaces. Then, applying it, we complete the theory of the spherical functions on the space X of nondegenerate unramified hermitian forms on a \mathfrak{p}-adic number field. More precisely, we give an explicit expression for the spherical functions, prove theorems on the spherical Fourier transforms on the space of Schwartz-Bruhat functions on X, and parametrize of all spherical functions on X. Finally, as an application, we give explicit expressions of local densities of representations of hermitian forms.