Harmonic analysis on the space of $p$-adic unitary hermitian matrices, including dyadic case

Abstract

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 size $m$ over a ${\mathfrak p}$-adic field $k$ and give unified description including dyadic case, which is a continuation of our previous papers on non-dyadic case. The space becomes complicated when $e = v_\pi(2) > 0$. First we introduce a typical spherical function $\omega(x;z)$ on $X$, and study their functional equations, which depend on $m$ and $e$, we give an explicit formula for $\omega(x;z)$, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as $m = 2n$ or $2n+1$, but independent of $e$. By spherical transform, we show the Schwartz space ${\mathcal S}(K \backslash X)$ is a free Hecke algebra ${\mathcal H}(G,K)$-module of rank $2^n$, and give parametrization of all the spherical functions on $X$ and the explicit Plancherel formula on ${\mathcal S}(K \backslash X)$. The Plancherel measure does not depend on $e$, but the normalization of $G$-invariant measure on $X$ depends.