Spherical functions of Hermitian and symmetric forms. III
type: Item
instance of: scholarly article
title: Spherical functions of Hermitian and symmetric forms. III
zbMATH Open document ID: 0674.43007
author: Yumiko Hironaka
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: [For part I see the preceding review Zbl 0674.43006.] This paper is the third part of a series of papers, whose aim is to show the functional equation for spherical functions and determine their possible poles for the space of nondegenerate Hermitian, or symmetric matrices over a p-adic number field. With the notation of part I the following theorem is proved. (1) Let \({\mathcal S}_ n\) be the symmetric group on n letters. For each \(\sigma\in {\mathcal S}_ n\), there exists a matrix C(\(\sigma\),z) in \(GL_{4^ n}({\mathbb{C}}(q^{z_ 1}...,q^{z_ n}))\) such that \[ L(x;\chi;\sigma (z))=\sum_{\chi}C(\sigma,z)_{\chi,\chi '}L(x,\chi ',z). \] (2) The function \(\prod_{1\leq i,j\leq n,i\neq j}(q^{2z_ i}-q^{2z_ j- 1})L(x;\chi;z)\) is a polynomial in \(q^{\pm z_ 1},...,q^{\pm z_ n}.\)
reviewed by: Masakazu Muro
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 4103693
zbMATH Keywords: Hermitian forms
zbMATH Keywords: functional equation
zbMATH Keywords: spherical functions
zbMATH Keywords: poles
zbMATH Keywords: symmetric matrices
zbMATH Keywords: p-adic number field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227929
OpenAlex ID: W1973990660
cites work: Rationality criteria and application to the generating series of a system of equations with coefficients in a local field
cites work: Q3828486
cites work: Q3200019
cites work: Eigenspaces of invariant differential operators on an affine symmetric space
DOI: 10.2748/TMJ/1178227929
description: scientific article; zbMATH DE number 4103693