Transfer of \(R\)-groups between \(p\)-adic inner forms of \(\mathrm{SL}_n\)
type: Item
instance of: scholarly article
title: Transfer of \(R\)-groups between \(p\)-adic inner forms of \(\mathrm{SL}_n\)
zbMATH Open document ID: 1318.22007
author: Kwangho Choiy
author: David Goldberg
published in: Manuscripta Mathematica
publication date: 2015-01-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.5054
review text: The study of R-groups is crucial to the understanding of the elliptic tempered spectrum of reductive \(p\)-adic groups. The article focuses on the interesting case of the inner forms of the special linear group. Let \(G=\mathrm{SL}_n(F)\), with \(F\) a \(p\)-adic field of characteristic zero, and let \(G'=\mathrm{SL}_m(D)\), where \(D\) is a central division algebra of degree \(d^2\) over \(F\), and \(md=n\). Let \(\mathrm{tG}=\mathrm{GL}_n(F)\) and \(\mathrm{tG}'=\mathrm{GL}_m(D)\). Let \(\sigma\) be an irreducible discrete series representation of a Levi subgroup \(M\) of \(G\), and let \(W(\sigma)\) denote its stabilizer in the Weyl group of \(M\). We denote by \(W_\sigma'\) the subgroup of \(W(\sigma)\) generated by the root reflections in the zeros of the rank-one Plancherel measures. The Knapp-Stein (or analytic) R-group of \(\sigma\) can be realized as the quotient of \(W(\sigma)\) by \(W'_\sigma\). Let \(\sigma\) be an irreducible discrete series representation of a Levi subgroup \(tM\) of \(\mathrm{tG}\) such that \(\sigma\) occurs in the restriction of \(\sigma\) to \(M=tM\cap G\). We will use analogous notation for the corresponding data attached to \(G'\) instead of \(G\). If \(\sigma\) and \(\sigma'\) correspond under the Jacquet-Langlands correspondence then we have \(W'_\sigma=W'_{\sigma'}\). The inclusion \(W(\sigma')\hookrightarrow W(\sigma)\) induces an inclusion \(R_\sigma\hookrightarrow R_{\sigma'}\). The authors provide a characterization of the quotient \(R_{\sigma'}/R_{\sigma'}\) in Remark 4.21 of the article. Important results of the article (Theorem 4.10 and Theorem 4.16, respectively) show that \(R_{\sigma}\) and \(R_{\sigma'}\) are isomorphic to the Arthur (or geometric) R-group of \(\sigma\) and \(\sigma'\), respectively. Moreover, Theorem 4.19 proves that all the irreducible discrete series representations belonging to a given L-packet of \(G'\) admit the same R-group. The similar result for \(G\) was proved previously by the second auth…
reviewed by: Anne-Marie Aubert
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 6387831
zbMATH Keywords: R-group
zbMATH Keywords: \(p\)-adic groups
zbMATH Keywords: inner forms of the special linear group
zbMATH Keywords: representations
MaRDI profile type: Publication
OpenAlex ID: W2077805188
arXiv ID: 1211.5054
cites work: On elliptic tempered characters
cites work: A note on \(L\)-packets
cites work: The endoscopic classification of representations. Orthogonal and symplectic groups
cites work: Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations. With an appendix by Neven Grbac.
cites work: Q5481252
cites work: Duality and the normalization of standard intertwining operators
cites work: Arthur R-groups, classical R-groups, and Aubert involutions for SO(2n + 1)
cites work: Stable trace formula: Cuspidal tempered terms
cites work: Induced representations of reductive ${\germ p}$-adic groups. I
cites work: Dual \(R\)-groups of the inner form of \(\mathrm{SL}(N)\)
cites work: Transfer of Plancherel Measures for Unitary Supercuspidal Representations betweenp-adic Inner Forms
cites work: On multiplicity in restriction of tempered representations of \(p\)-adic groups
cites work: Q3708223
cites work: L-indistinguishability and R groups for the special linear group
cites work: \(R\)-groups and elliptic representations for \(SL_ n\)
cites work: Reducibility of Induced Representations for SP(2N) and SO(N)
cites work: Reducibility for SUn and Generic Elliptic Representations
cites work: The local Langlands conjecture for \(\text{Sp}(4)\)
cites work: A submersion principle and its applications
cites work: Représentations du groupe de Weil d'un corps local
cites work: A simple proof of Langlands conjectures for \(\text{GL}_n\) on a \(p\)-adic field
cites work: The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich
cites work: Q4361592
cites work: Q5662173
cites work: Q4297086
cites work: Q3726219
cites work: On the Cohomology of Certain Non-Compact Shimura Varieties (AM-173)
cites work: Q4692288
cites work: Q3635891
cites work: Representations of \(GL(n)\) and division algebras over a p-adic field
cites work: Some Results on L-Indistinguishability for SL(r)
cites work: Q4012130
cites work: The Knapp-Stein Dimension Theorem for p-Adic Groups
cites work: Notes on representation theory of non-Archimedean \(SL(n)\)
cites work: Exercices dyadiques
DOI: 10.1007/S00229-014-0689-3
description: scientific article