TY - JOUR

T1 - Spherical functions on the space of p-adic unitary hermitian matrices

AU - Hironaka, Yumiko

AU - Komori, Yasushi

N1 - Funding Information:
This research was partially supported by Grant-in-Aid for scientific Research (C): 22540045, 24540031, 25400026. The authors would like to express their thanks to the reviewers for their careful reading of the manuscript.

PY - 2014/3

Y1 - 2014/3

N2 - We investigate the space X of unitary hermitian matrices over 𝔭-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 ∉ 𝔭. 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 $\mathcal{S}(K{\backslash}X)$.

AB - We investigate the space X of unitary hermitian matrices over 𝔭-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 ∉ 𝔭. 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 $\mathcal{S}(K{\backslash}X)$.

KW - Hall-Littlewood symmetric polynomials

KW - Plancherel formula

KW - Spherical functions

KW - hermitian matrices

KW - unitary groups

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U2 - 10.1142/S1793042113501066

DO - 10.1142/S1793042113501066

M3 - Article

AN - SCOPUS:84894676107

VL - 10

SP - 513

EP - 558

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 2

ER -