TY - JOUR

T1 - Harmonic analysis on the space of p-adic unitary hermitian matrices, mainly for dyadic case

AU - Hironaka, Yumiko

PY - 2017/12

Y1 - 2017/12

N2 - 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 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 ω(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 Cn 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 S(K\X) is a free Hecke algebra H(G, K)-module of rank 2n, and give parametrization of all the spherical functions on X and the explicit Plancherel formula on S(K\X). The Plancherel measure does not depend on e, but the normalization of G-invariant measure on X depends.

AB - 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 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 ω(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 Cn 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 S(K\X) is a free Hecke algebra H(G, K)-module of rank 2n, and give parametrization of all the spherical functions on X and the explicit Plancherel formula on S(K\X). The Plancherel measure does not depend on e, but the normalization of G-invariant measure on X depends.

KW - Dyadic fields

KW - Hall-Littlewood polynomials

KW - Hermitian matrices

KW - Plancherel formula

KW - Spherical functions

KW - Unitary groups

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U2 - 10.3836/tjm/1502179240

DO - 10.3836/tjm/1502179240

M3 - Article

AN - SCOPUS:85040446279

VL - 40

SP - 517

EP - 564

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 2

ER -