Spherical functions and local densities on Hermitian forms
type: Item
instance of: scholarly article
title: Spherical functions and local densities on Hermitian forms
zbMATH Open document ID: 0936.11024
author: Yumiko Hironaka
published in: Journal of the Mathematical Society of Japan
publication date: 2000-05-24T00:00:00Z
review text: The aim of this paper is to complete the theory of the spherical functions on the space \(X\) of unramified hermitian forms over a \({\mathfrak p}\)-adic field \(k\) [Jap. J. Math., New Ser. 14, 203-223 (1988; Zbl 0674.43006), ibid. 15, 15-51 (1989; Zbl 0714.43012), Tôhoku Math. J., II. Ser. 40, 651-671 (1988; Zbl 0674.43007), Comment. Math. Univ. St. Pauli 39, 157-193 (1990; Zbl 0718.11016)]. A nonzero \(K\)-invariant function on \(X\) is called a spherical function on \(X\) if it is a common eigenfunction under the action of the Hecke algebra \({\mathcal H} (G,K)\) by the convolution product, where \(G= GL_n(k)\) and \(K= GL_n ({\mathcal O}_k)\). The author gives an explicit expression for the typical spherical functions and defines the spherical Fourier transform on the space \({\mathcal S}(K\setminus X)\) of \(K\)-invariant Schwartz-Bruhat functions on \(X\). The structure of \({\mathcal S}(K\setminus X)\) as \({\mathcal H}(G,K)\)-modules, the Plancherel formula and the inversion formula of the spherical Fourier transform, and a parametrization of all spherical functions on \(X\) are given explicitly. Finally, as an application of the theory, the author gives explicit expressions of local densities of integral representations of hermitian forms.
reviewed by: Yumiko Hironaka
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 1347249
zbMATH Keywords: local field
zbMATH Keywords: hermitian forms
zbMATH Keywords: spherical Fourier transform
zbMATH Keywords: \(K\)-invariant Schwartz-Bruhat functions
zbMATH Keywords: Plancherel formula
zbMATH Keywords: inversion formula
zbMATH Keywords: local densities of integral representations of hermitian forms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05130553
OpenAlex ID: W1984781807
DOI: 10.2969/JMSJ/05130553
description: scientific article; zbMATH DE number 1347249