The orbit and \(\theta\) correspondence for some dual pairs
type: Item
instance of: scholarly article
title: The orbit and \(\theta\) correspondence for some dual pairs
zbMATH Open document ID: 0848.22023
author: Steve Rallis
author: Gérard Schiffmann
published in: Journal of Mathematics of Kyoto University
publication date: 1996-10-31T00:00:00Z
review text: Let \(G\) be a semisimple Lie group and \(H\), \(H'\) reductive subgroups, each being the full centralizer of the other. Let \(\omega\) be an irreducible unitary representation of \(G\) which is minimal in the sense that \(\omega\) corresponds to a coadjoint orbit of minimal dimension. In the case \(G= Sp_n\) and \(\omega\) being the Weil representation, \textit{R. Howe} has shown [J. Am. Math. Soc. 2, 535-552 (1989; Zbl 0716.22006)] that there is a homomorphism \(\psi: {\mathfrak z}\to {\mathfrak z}'\) of the center of the universal enveloping algebras such that on the smooth vectors we have \(\omega (z)= \omega (\psi (z))\) for all \(z\in {\mathfrak z}\). This is part of a more general study of the so called Howe correspondence, the representation theoretic gadget underlying the theta correspondence in the theory of automorphic forms. -- In the present paper this is generalized to simple \(G\) not of type \(A_n\) and certain standard pairs \(H\), \(H'\) which are given by the components of the Dynkin diagram when one removes a simple root. Although it only treats very special dual pairs the paper is an important step towards the generalization of Howe theory to groups other than the symplectic one.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 880139
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: unitary representation
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: Weil representation
zbMATH Keywords: universal enveloping algebras
zbMATH Keywords: Howe correspondence
zbMATH Keywords: automorphic forms
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: dual pairs
zbMATH Keywords: Howe theory
reviewed by: Anton Deitmar
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518706
OpenAlex ID: W1563017432
DOI: 10.1215/KJM/1250518706
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description: scientific article; zbMATH DE number 880139