Harmonic analysis on the finite symmetric space \(\mathrm{GL}(n,K)/\mathrm{GL}(1,K)\times \mathrm{GL}(n-1,K)\)
type: Item
instance of: scholarly article
title: Harmonic analysis on the finite symmetric space \(\mathrm{GL}(n,K)/\mathrm{GL}(1,K)\times \mathrm{GL}(n-1,K)\)
zbMATH Open document ID: 0835.43012
DOI: 10.1016/0019-3577(95)91240-V
published in: Indagationes Mathematicae. New Series
publication date: 1995-10-09T00:00:00Z
review text: This paper gives a classification of all irreducible spherical representations of the group \(G = \mathrm{GL}(n,K)\) with respect to the subgroup \(H = \mathrm{GL}(1,K) \times \mathrm{GL}(n - 1,K)\). Here \(K\) denotes a finite field of odd characteristic. The spherical functions associated to these representations are also computed. Among other results, the author shows that \((G,H)\) is not a Gelfand pair and there are no irreducible spherical cuspidal representations for \(n > 2\). Cusp forms and cuspidal representations play an important role in proofs following some ideas of \textit{Harish-Chandra} [Collected Papers. Vol. IV, Springer, Berlin, 8--20 (1984)]. As the author mentions, this work may be considered as a particular case of some results of \textit{G. Lusztig} [Prog. Math. 88, 57--81 (1990; Zbl 0773.20012)]. These ideas may be used in related matters of harmonic analysis on finite symmetric spaces.
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 792822
zbMATH Keywords: cusp forms
zbMATH Keywords: irreducible spherical representations
zbMATH Keywords: spherical functions
zbMATH Keywords: Gelfand pair
zbMATH Keywords: irreducible spherical cuspidal representations
author: Gerrit van Dijk
MaRDI profile type: Publication
cites work: Q3682663
cites work: Q3211549
cites work: Eisenstein Series over Finite Fields
cites work: Q3972765
cites work: Complex Representations of 𝐺𝐿(2,𝐾) for Finite Fields 𝐾
cites work: Q5181335
full work available at URL: 0019-3577(95)91240-v
OpenAlex ID: W2048263240
description: scientific article; zbMATH DE number 792822