Spherical functions and local densities on hermitian forms

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.