Spherical functions and local densities on the space of p -adic quaternion Hermitian forms

Yumiko Hironaka*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We introduce the space X of quaternion Hermitian forms of size n on a p-adic field with odd residual characteristic, and define typical spherical functions ω(x; s) on X and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to Sn, and define a spherical Fourier transform on the Schwartz space S(K\X) which is Hecke algebra ℋ(G,K)-injective map into the symmetric Laurent polynomial ring of size n. Then, we determine the explicit formulas of ω(x; s) by a method of the author's former result. In the last section, we give precise generators of (K\X) and determine all the spherical functions for n ≤ 4, and give the Plancherel formula for n = 2.

Original languageEnglish
JournalInternational Journal of Number Theory
DOIs
Publication statusAccepted/In press - 2021

Keywords

  • orthogonal polynomials
  • Plancherel formula
  • quaternion Hermitian forms
  • Spherical function

ASJC Scopus subject areas

  • Algebra and Number Theory

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