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December 2017 Harmonic Analysis on the Space of $p$-adic Unitary Hermitian Matrices, Mainly for Dyadic Case
Yumiko HIRONAKA
Tokyo J. Math. 40(2): 517-564 (December 2017). DOI: 10.3836/tjm/1502179240

Abstract

We are interested in 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 $k$ mainly for dyadic case, and give the unified description with our previous papers for non-dyadic case. The space becomes complicated for dyadic case, and the set of integral elements in $X$ has plural Cartan orbits. We introduce a typical spherical function $\omega(x;z)$ on $X$, study its functional equations, which depend on $m$ and the ramification index $e$ of $2$ in $k$, and give its explicit formula, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as the parity $m = 2n$ 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.

Citation

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Yumiko HIRONAKA. "Harmonic Analysis on the Space of $p$-adic Unitary Hermitian Matrices, Mainly for Dyadic Case." Tokyo J. Math. 40 (2) 517 - 564, December 2017. https://doi.org/10.3836/tjm/1502179240

Information

Published: December 2017
First available in Project Euclid: 9 January 2018

zbMATH: 06855947
MathSciNet: MR3743731
Digital Object Identifier: 10.3836/tjm/1502179240

Subjects:
Primary: 11E85
Secondary: 11E95, 11F70, 22E50, 33D52

Rights: Copyright © 2017 Publication Committee for the Tokyo Journal of Mathematics

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Vol.40 • No. 2 • December 2017
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