Spherical functions on a finite affine space with a series of Zelevinsky subgroups
type: Item
instance of: scholarly article
title: Spherical functions on a finite affine space with a series of Zelevinsky subgroups
zbMATH Open document ID: 0878.43019
DOI: 10.1007/BF02303977
author: Egor E. Petrov
published in: Mathematical Notes
publication date: 1996-02-21T00:00:00Z
review text: In the complete linear group \(G_n= GL(n,F)\) over a finite field \(F\) the Zelevinsky series [cf. \textit{A. V. Zelevinsky}, Representations of finite classical groups. A Hopf algebra approach. Lect. Notes Math. 869 (Berlin etc. 1981; Zbl 0465.20009)] \(G_{n-1/2} \supset G_{n-1} \supset\cdots \supset G_{3/2} \supset G_1\) of subgroups \(G_{n-i +1/2}\), \(G_{n-i}\), \(i=1,2, \dots, n-1\), is given. The series generates a simple branching quasiregular representation of the group \(G_n\) associated with the homogeneous space \(G_n/G_{n-1/2}\). The spherical subgroup \(G_{n-1/2}\) separates the subspace of spherical functions, and the series introduces a Gel'fand-Tsetlin basis of basic spherical functions. In the paper the explicit form of these spherical functions is found.
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 846503
zbMATH Keywords: linear group
zbMATH Keywords: Zelevinsky series
zbMATH Keywords: quasiregular representation
zbMATH Keywords: homogeneous space
zbMATH Keywords: spherical functions
reviewed by: Kaarin Riives
MaRDI profile type: Publication
cites work: Representations of finite classical groups. A Hopf algebra approach
cites work: Q3803623
description: scientific article; zbMATH DE number 846503