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

Yumiko Hironaka, Yasushi Komori

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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
Issue number2
Publication statusPublished - 2014 Mar


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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