Paquets stables de représentations tempérées et de réduction unipotente pour \(\text{SO}(2n+1)\). (Stable packets of tempered representations and unipotent reduction for \(\text{SO}(2n+1)\).)
type: Item
instance of: scholarly article
title: Paquets stables de représentations tempérées et de réduction unipotente pour \(\text{SO}(2n+1)\). (Stable packets of tempered representations and unipotent reduction for \(\text{SO}(2n+1)\).)
zbMATH Open document ID: 1037.22036
author: Jean-Loup Waldspurger
author: Colette Moeglin
published in: Inventiones Mathematicae
publication date: 2003-07-08T00:00:00Z
review text: Let \(p\) be a prime number, and let \(F\) be a finite extension of \(\mathbb Q_p\). Let \(\mathbf G_{iso}\) (resp. \(\mathbf G_{an}\)) denote the special orthogonal group of a form with anisotropic kernel of dimension \(1\) (resp. \(3\)). Hence the group \(\mathbf G_{iso}\) is \(F\)-split, and the group \(\mathbf G_{an}\) is a non-split inner form of \(\mathbf G_{iso}\). For \(\#\in\{iso,an\}\), let \(G_{\#}\) be the group \(\mathbf G_{\#}(F)\) of the \(F\)-points of \(\mathbf G_{\#}\). Let \(\text{Irr}_u^{G_{\#}}\) denote the set isomorphism classes of irreducible admissible representations of \(G_{\#}\) with \textit{unipotent reduction}, that is, the subset of the set of isomorphism classes of level zero irreducible admissible representations of \(G_{\#}\) such that the corresponding cuspidal representation of the reductive quotient (a finite group of Lie type) of the parahoric subgroup is \textit{unipotent}. Lusztig has proved that this set is in bijective correspondence with the set \(\underline\Psi_u^{\#}\) of all \(\text{Sp}(2n,\mathbb C)\)-conjugacy classes of pairs \((\psi,\varepsilon)\) with \(\psi\colon W_F\times\text{SL}(2,\mathbb C)\to\text{Sp}(2n,\mathbb C)\) an homomorphism such that its restriction to \(W_F\) (the Weil group of \(F\)) is semisimple and trivial on the inertia subgroup of \(W_F\), and its restriction to \(\text{SL}(2,\mathbb C)\) is algebraic, and \(\epsilon\colon\,A(\psi)=\mathbb Z_{\text{Sp}(2n,\mathbb C)}(\psi)/\mathbb Z_{\text{Sp}(2n,\mathbb C)}^0(\psi)\to\{\pm1\}\) is a character such that its composition with the natural homomorphism from the centre of \(\text{Sp}(2n,\mathbb C)\) to \(A(\psi)\) is equal to \(1\) (resp. \(-1\)) if \(\#=iso\) (resp. \(an\)). (Here \(\mathbb Z_{\text{Sp}(2n,\mathbb C)}(\psi)\) denotes the centralizer of the image of \(\psi\), and \(\mathbb Z_{\text{Sp}(2n,\mathbb C)}^0(\psi)\) the connected component of the latter.) The authors consider the subset \(\text{Irr}_{utemp}^{G_{\#}}\) of the isomorphism c…
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1947265
zbMATH Keywords: tempered representation
zbMATH Keywords: \(p\)-adic classical group
zbMATH Keywords: level zero
zbMATH Keywords: stable packet
zbMATH Keywords: endoscopic transfert
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: s00222-002-0274-3
OpenAlex ID: W2563078116
DOI: 10.1007/S00222-002-0274-3
description: scientific article; zbMATH DE number 1947265