Irreducibility of standard representations for Iwahori-spherical representations.
type: Item
instance of: scholarly article
title: Irreducibility of standard representations for Iwahori-spherical representations.
zbMATH Open document ID: 0910.22018
author: Goran Muić
author: Freydoon Shahidi
published in: Mathematische Annalen
publication date: 1999-04-15T00:00:00Z
review text: Let \(\mathbb G\) be a split connected reductive group over a local nonarchimedean field \(F\) of characteristic zero, and let \(\mathbb B = \mathbb T \mathbb U\) be a Borel subgroup of \(\mathbb G\), where \(\mathbb T\) is a maximal split torus of \(\mathbb B\) and \(\mathbb U\) is its unipotent radical. Let \(\mathbb P = \mathbb M \mathbb N\) be a Levi decomposition of a parabolic subgroup \(\mathbb P\) of \(\mathbb G\) with \(\mathbb T \subset \mathbb M\) and \(\mathbb N \subset \mathbb U\), and let \(\sigma\) be an Iwahori-spherical tempered generic representation of \(M = \mathbb M (F)\). Fix an element \(\nu\) of the complex dual \(\mathfrak a^\ast_{\mathbb C}\) of the real Lie algebra of the split component \(\mathbb A\) of \(\mathbb M\) in the positive Weyl chamber, and denote by \(I(\nu,\sigma)\) the standard representation induced from \(\nu\) and \(\sigma\). In this paper the authors prove that \(I(\nu, \sigma)\) is irreducible if and only if the Langlands quotient \(J(\nu, \sigma)\) of \(I(\nu, \sigma)\) is generic. They also show that points of reducibility can be computed explicitly.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1221879
zbMATH Keywords: Iwahori-spherical representations
zbMATH Keywords: tempered representations
zbMATH Keywords: principal series
zbMATH Keywords: reductive groups
MaRDI profile type: Publication
full work available at URL: s002080050216
OpenAlex ID: W2049069769
DOI: 10.1007/S002080050216
description: scientific article; zbMATH DE number 1221879