Spherical Functions on the Space of p-adic Unitary Hermitian Matrices II, the Case of Odd Size

We are interested in the harmonic analysis on p-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space X of unitary hermitian matrices of odd size over a p-adic field of odd residual characteristic, which is a continuation of our previous paper where we have studied for even size matrices. First we give the explicit representatives of the Cartan decomposition of X and introduce a typical spherical function (x; z) on X. After studying the functional equations, we give an explicit formula for (x; z), where Hall-Littlewood polynomials of type C n appear as a main term, though the unitary group acting on X is of type BC n. By spherical transform, we show the Schwartz space S (K) is a free Hecke algebra H (G, K) -module of rank 2 n, where 2n + 1 is the size of matrices in X, and give parametrization of all the spherical functions on X and the explicit Plancherel formula on S (K).