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 Cn 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 2n, 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