The oscillator correspondence of orbital integrals, for pairs of type one in the stable range
type: Item
instance of: scholarly article
title: The oscillator correspondence of orbital integrals, for pairs of type one in the stable range
zbMATH Open document ID: 0859.22008
DOI: 10.1215/S0012-7094-96-08201-0
author: Andrzej Daszkiewicz
author: Tomasz Przebinda
published in: Duke Mathematical Journal
publication date: 1996-09-05T00:00:00Z
review text: Let \((G,G') \subseteq \text{Sp}(W)\) be a real reductive dual pair in the sense of Howe. Howe duality defines a bijection \(\Pi \leftrightarrow \Pi'\) between certain sets of irreducible, admissible representations of \(G\) and \(G'\), or more precisely, of their twofold coverings in the metaplectic group. One of the fundamental questions in Howe duality theory is how various properties and invariants of \(\Pi\) and \(\Pi'\) are related. Howe has conjectured that for generic \(\Pi\) and \(\Pi'\) the wave front sets \(WF(\Pi)\) and \(WF(\Pi')\) should be related via the moment maps \(\tau:W\to{\mathfrak g}^*\), \(\tau':W\to{\mathfrak g}'{}^*\) as follows: \(WF(\Pi')=\tau'(\tau^{-1}(WF(\Pi)))\). For pairs of type I in the stable range the authors describe a correspondence of orbital integrals on \(\mathfrak g\) and \({\mathfrak g}'\) which agrees with the correspondence of orbits defined by these maps, via an explicit integral kernel operator. This allows a more precise formulation of Howe's conjecture by invoking the Fourier transform and an asymptotic expansion of characters. Under some strong assumptions (``deep stable range'') the formula \(WF(\Pi')=\tau'(\tau^{-1}(WF(\Pi)))\) was proved by the authors in [Invent. Math. 123, 349-376 (1996; Zbl 0845.22007)].
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 908679
zbMATH Keywords: dual pair
zbMATH Keywords: Howe duality
zbMATH Keywords: representations
zbMATH Keywords: wave front sets
zbMATH Keywords: orbital integrals
zbMATH Keywords: Fourier transform
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08201-0
OpenAlex ID: W1568643699
description: scientific article; zbMATH DE number 908679