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

TL;DR

This paper studies spherical functions on the space of p-adic unitary hermitian matrices, providing explicit formulas, parametrizations, and a Plancherel formula, revealing deep algebraic and harmonic analysis structures.

Contribution

It offers explicit formulas for spherical functions on p-adic unitary hermitian matrices and characterizes the Schwartz space as a free Hecke algebra module.

Findings

Explicit spherical function formulas involving Hall-Littlewood polynomials

Parametrization of all spherical functions on the space

Plancherel formula for the Schwartz space

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 $\SKX$.