Spherical functions on the space of $p$-adic unitary hermitian matrices

Abstract

We investigate the space $X$ of unitary hermitian matrices over \frp-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic, i.e., 2\notin \frp. In the latter half we assume odd residual characteristic, and give explicit formulas of typical spherical functions on $X$, where Hall-Littlewood symmetric polynomials of type $C_n$ appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show the Schwartz space \SKX is a free Hecke algebra \hec-module of rank $2^n$, where $2n$ is the size of matrices in $X$, and give the explicit Plancherel formula on \SKX.Comment: to appear in International Journal of Number Theor