### Abstract

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 C_{n} 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 2^{n}, where 2n is the size of matrices in X, and give the explicit Plancherel formula on $\mathcal{S}(K{\backslash}X)$.

Original language | English |
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Pages (from-to) | 513-558 |

Number of pages | 46 |

Journal | International Journal of Number Theory |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Mar 1 |

### Keywords

- Hall-Littlewood symmetric polynomials
- Plancherel formula
- Spherical functions
- hermitian matrices
- unitary groups

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*International Journal of Number Theory*,

*10*(2), 513-558. https://doi.org/10.1142/S1793042113501066