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

Yumiko Hironaka, Yasushi Komori

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    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 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 $\mathcal{S}(K{\backslash}X)$.

    Original languageEnglish
    Pages (from-to)513-558
    Number of pages46
    JournalInternational Journal of Number Theory
    Volume10
    Issue number2
    DOIs
    Publication statusPublished - 2014 Mar

    Fingerprint

    Spherical Functions
    Unitary matrix
    Hermitian matrix
    P-adic
    Explicit Formula
    Odd
    Plancherel Formula
    Schwartz Space
    Symmetric Polynomials
    Free Algebras
    Hecke Algebra
    Parametrization
    Fourier transform
    Decompose
    Module
    Term

    Keywords

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

    ASJC Scopus subject areas

    • Algebra and Number Theory

    Cite this

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

    In: International Journal of Number Theory, Vol. 10, No. 2, 03.2014, p. 513-558.

    Research output: Contribution to journalArticle

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