### 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 |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*International Journal of Number Theory*,

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

**Spherical functions on the space of p-adic unitary hermitian matrices.** / Hironaka, Yumiko; Komori, Yasushi.

Research output: Contribution to journal › Article

*International Journal of Number Theory*, vol. 10, no. 2, pp. 513-558. https://doi.org/10.1142/S1793042113501066

}

TY - JOUR

T1 - Spherical functions on the space of p-adic unitary hermitian matrices

AU - Hironaka, Yumiko

AU - Komori, Yasushi

PY - 2014/3

Y1 - 2014/3

N2 - 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 Cn 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 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on $\mathcal{S}(K{\backslash}X)$.

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

KW - Hall-Littlewood symmetric polynomials

KW - hermitian matrices

KW - Plancherel formula

KW - Spherical functions

KW - unitary groups

UR - http://www.scopus.com/inward/record.url?scp=84894676107&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894676107&partnerID=8YFLogxK

U2 - 10.1142/S1793042113501066

DO - 10.1142/S1793042113501066

M3 - Article

AN - SCOPUS:84894676107

VL - 10

SP - 513

EP - 558

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 2

ER -