The dual pair (U(3),U(1)) over a p-adic field
type: Item
instance of: scholarly article
title: The dual pair (U(3),U(1)) over a p-adic field
zbMATH Open document ID: 0675.22008
author: Courtney Moen
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: This paper treats some interesting aspects of Howe's correspondence for the dual reductive pair (U(3),U(1)) over a p-adic field, with p odd. Roughly speaking, the idea is this: one embeds the group U(3) (with U(1) embedded as the center) inside the rank 3 symplectic group \(Sp_ 3\), and considers the restriction of the canonical metaplectic representation \(\omega_{\psi}\) of \(Sp_ 3\) to this unitary subgroup; according to Howe's general theory, we have a decomposition \[ \omega_{\psi}|_{U(3)}=\oplus_{\chi}\pi_{\chi} \] where \(\chi\) runs through the character group of U(1) (which is isomorphic to the norm 1 group of the quadratic extension E over which U(1) and U(3) are defined), and \(\pi_{\chi}\) is an irreducible admissible representation of U(3). The problem considered in this paper is to describe \(\pi_{\chi}\) as explicitly as possible, in terms of \(\chi\). First it is shown that \(\pi_{\chi}\) is supercuspidal (not a constituent of any induced representation ind \(\nu)\) if and only if \(\chi\) is ``ramified'' with respect to the additive character \(\psi\) defining \(\omega_{\psi}\) (the precise condition is that \(\chi\) should not occur in the restriction to U(1) - embedded as a compact Cartan subgroup - of the canonical Weil representation \(\omega^ 1_{\psi}\) of \(SL_ 2(F))\). In case \(\pi_{\chi}\) is not supercuspidal, the author gives an explicit description of the character \(\nu^*\) of \(E^ x\) such that \(\pi_{\chi}\) is an irreducible constituent of Ind \(\nu^*| |^{1/2}\). On the other hand, if \(\pi_{\chi}\) is supercuspidal, then the author explicitly constructs some representation \(\sigma\) of some maximal compact subgroup of U(3) such that \(\pi_{\chi}=ind \sigma\); here it seems that the group over the ring of integers does not suffice - in almost all cases of E and \(\chi\), the other class of maximal compact subgroups is required. Reviewer's Remarks. The Weil representation \(\omega_{\psi}|_{U(3)}\) is actually obtained in two states. …
reviewed by: Stephen Gelbart
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 4105259
zbMATH Keywords: supercuspidal representations
zbMATH Keywords: Howe's correspondence
zbMATH Keywords: dual reductive pair
zbMATH Keywords: p-adic field
zbMATH Keywords: symplectic group
zbMATH Keywords: metaplectic representation
zbMATH Keywords: character group
zbMATH Keywords: irreducible admissible representation
zbMATH Keywords: irreducible constituent
zbMATH Keywords: maximal compact subgroups
zbMATH Keywords: Weil representation
zbMATH Keywords: principal series
MaRDI profile type: Publication
full work available at URL: pjm.1987.127.141
OpenAlex ID: W1572339533
DOI: 10.2140/PJM.1987.127.141
Recommended article: The dual pair \((U(1),U(1))\) over a \(p\)-adic field
Recommended article: The Dual Pair G2 × PU3 (D) (p-Adic Case)
Recommended article: The unitary dual of \(p\)-adic \(G_2\)
Recommended article: Q4314329
Recommended article: Representations of U (2,1) over a p-adic field.
Recommended article: The \(u\)-invariant of \(p\)-adic function fields
Recommended article: The unitary dual of \(\mathrm{GL}(n)\) over an archimedean field
Recommended article: Q3165609
Recommended article: Unipotent Representations and Reductive Dual Pairs over Finite Fields
Recommended article: On theU-Invariant ofP-Adic Function Fields
description: scientific article; zbMATH DE number 4105259