### 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 |
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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)