### 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 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 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 S(K\X) is a free Hecke algebra H(G, K)-module of rank 2^{n}, 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.

Original language | English |
---|---|

Pages (from-to) | 517-564 |

Number of pages | 48 |

Journal | Tokyo Journal of Mathematics |

Volume | 40 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 Dec 1 |

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### Keywords

- Dyadic fields
- Hall-Littlewood polynomials
- Hermitian matrices
- Plancherel formula
- Spherical functions
- Unitary groups

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Harmonic analysis on the space of p-adic unitary hermitian matrices, mainly for dyadic case.** / Hironaka, Yumiko.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 40, no. 2, pp. 517-564. https://doi.org/10.3836/tjm/1502179240

}

TY - JOUR

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

AU - Hironaka, Yumiko

PY - 2017/12/1

Y1 - 2017/12/1

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

UR - http://www.scopus.com/inward/record.url?scp=85040446279&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040446279&partnerID=8YFLogxK

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 -