Supercuspidal representations and preservation principle of theta correspondence
type: Item
instance of: scholarly article
title: Supercuspidal representations and preservation principle of theta correspondence
zbMATH Open document ID: 1466.22003
author: Shu-Yen Pan
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2019-05-06T00:00:00Z
review text: The paper studies the``first occurrence of supercuspidal representations'' in relation with the local theta correspondence (also known as the Howe correspondence) for representations of reductive groups over \(p\)-adic fields. The main themes are the connections between the general case and the depth-zero case, and the interplay with the situation for finite fields. Before we can state the results more precisely, we must establish notation for the Howe correspondence. Let \(F\) be a \(p\)-adic field, and let \(D\) be either \(F\) or a quadratic extension of \(F\). Let \((G(V), G(V'))\) be a reductive dual pair attached to a pair or Hermitian or anti-Hermitian spaces over \(D\). More precisely, fix \(\epsilon = \pm 1\), let \(V\) (resp. \(V'\)) be a finite-dimensional nondegenerate \(\epsilon\)-Hermitian (resp. \((-\epsilon)\)-Hermitian) space over \(D\). Let \(G(V)\) and \(G(V')\) denote the corresponding groups of isometries, and let us assume that \((G(V), G'(V))\) is a reductive dual pair in \(\mathrm{Sp}(V \otimes_D V')\). Let \(\mathrm{Irr}(G(V))\) and \(\mathrm{Irr}(G(V'))\) denote the admissible duals of \(G(V)\) and \(G(V')\). The Howe correspondence determines a map \(\mathrm{Irr}(G(V)) \to \{0\} \cup \mathrm{Irr}(G(V'))\), which we will denote by \(\pi \to \theta(\pi, V')\). It is injective on the set of \(\pi \in \mathrm{Irr}(G(V))\) such that \(\pi \to \theta(\pi, V') \neq 0\). The author assumes \(p \neq 2\) throughout, in which case the correspondence is due to \textit{J. L. Waldspurger} [Isr. Math. Conf. Proc. 2, 267--324 (1990; Zbl 0722.22009)]. A central notion of the paper is that of ``first occurrence of supercuspidal representations'' in the correspondence. Let us describe it briefly. Fix an anisotropic \((-\epsilon)\)-Hermitian space \(V'_\mathrm{an}\), and for each nonnegative integer \(k\), fix a vector space \(V'_{k}\) given by the direct sum of \(k\) hyperbolic planes. Now let \(\pi\) be a supercuspidal irreducible representation of \(G(V)\)…
reviewed by: Alexandre Afgoustidis
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 7051099
zbMATH Keywords: representations of \(p\)-adic reductive groups
zbMATH Keywords: Theta correspondence
zbMATH Keywords: Howe correspondence
zbMATH Keywords: reductive dual pairs
zbMATH Keywords: depth-zero representations
MaRDI profile type: Publication
full work available at URL: crelle-2016-0050
OpenAlex ID: W2531361403
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DOI: 10.1515/CRELLE-2016-0050
description: scientific article; zbMATH DE number 7051099