On the irreducibility of some induced representations of real reductive Lie groups
type: Item
instance of: scholarly article
title: On the irreducibility of some induced representations of real reductive Lie groups
zbMATH Open document ID: 1471.22015
author: Atsushi Ichino
author: Wee Teck Gan
published in: Tunisian Journal of Mathematics
publication date: 2019-02-22T00:00:00Z
review text: In their earlier work on the Shimura-Waldspurger correpondence for \(\mathrm{Mp}_{\mathrm{2n}}\) [Ann. Math. (2) 188, No. 3, 965--1016 (2018; Zbl 1486.11076)], the authors use the irreducibility of certain induced representations of the metaplectic group \(\mathrm{Mp}_{2n}(F)\) and of the split orthogonal group \(\mathrm{SO}_{2n+1}(F)\), when \(F\) is a local field of characteristic zero. The paper under review establishes the necessary irreducibility theorems when \(F = \mathbb{R}\). When \(F \neq \mathbb{R}\), analogous irreducibility results follow from the work of \textit{C. Mœglin} [Clay Math. Proc. 13, 333--374 (2011; Zbl 1225.22015); Adv. Math. 228, No. 2, 1068--1134 (2011; Zbl 1225.22016)] for nonarchimedean \(F\), and [\textit{C. Moeglin} and \textit{D. Renard}, Contemp. Math. 691, 203--256 (2017; Zbl 1390.22013)] for \(F = \mathbb{C}\). The result for the metaplectic group \(\mathrm{Mp}_{2n}(\mathbb{R})\) (Proposition 2.4) is the irreducibility of \[ \mathrm{Ind}_{\widetilde{P}}^{\mathrm{Mp}_{2n}(\mathbb{R})}({\tau_\phi}\otimes \chi \otimes \pi_0) \] where \begin{itemize} \item \(\widetilde{P}\) is a parabolic subgroup of \(\mathrm{Mp}_{2n}(\mathbb{R})\) given by the lift to \(\mathrm{Mp}_{2n}(\mathbb{R})\) of a parabolic subgroup of \(\mathrm{Sp}_{2n}(\mathbb{R})\) whose Levi factor is isomorphic with \(\mathrm{GL}_{k}(\mathbb{R}) \times \mathrm{Sp}_{2n_0}(\mathbb{R})\), with \(k+n_0=n\). \item \(\phi: W_\mathbb{R} \to \mathrm{GL}_k(\mathbb{C})\) is a \(k\)-dimensional representation of the Weil group \(W_\mathbb{R}\) whose irreducible summands are all nonsymplectic and almost tempered, \item \(\tau_\phi\) is the irreducible representation of \(\mathrm{GL}_k(\mathbb{R})\) associated to \(\phi\) by the Langlands correspondence, \item \(\chi\) is a genuine character of the two-fold cover of \(\mathrm{GL}_k(\mathbb{R})\), \item \(\pi_0\) is a representation of \(\mathrm{Mp}_{2n_0}(\mathbb{R})\) in the \(L\)-packet attached to a Langlands parameter \(\psi_0: …
reviewed by: Alexandre Afgoustidis
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E47
zbMATH DE Number: 7027518
zbMATH Keywords: irreducibility
zbMATH Keywords: real reductive groups
zbMATH Keywords: metaplectic groups
zbMATH Keywords: Arthur packets
Wikidata QID: Q115230541
MaRDI profile type: Publication
OpenAlex ID: W2789963805
cites work: Q4209242
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DOI: 10.2140/TUNIS.2019.1.73
description: scientific article; zbMATH DE number 7027518