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

Yumiko Hironaka, Yasushi Komori

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1 Citation (Scopus)

## Abstract

We investigate the space X of unitary hermitian matrices over &pfr;-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 &NotElement; &pfr;. From Sec. 2.2 till the end of Sec. 4, 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 that the Schwartz space $\mathcal{S}(K{\backslash}X)$ is a free Hecke algebra $\mathcal{H}(G,K)$-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on $\mathcal{S}(K{\backslash}X)$.

Original language English 513-558 46 International Journal of Number Theory 10 2 https://doi.org/10.1142/S1793042113501066 Published - 2014 Mar

## Keywords

• Hall-Littlewood symmetric polynomials
• Plancherel formula
• Spherical functions
• hermitian matrices
• unitary groups

## ASJC Scopus subject areas

• Algebra and Number Theory