The wave front set correspondence for dual pairs with one member compact

2021/08/24 by McKee, M., Angela Pasquale, Pasquale, A. +1
Mathematics · #22E30 #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Mathematical Analysis and Transform Methods #Primary: 22E45 #Representation Theory (math.RT) #secondary: 22E46

paper · doi:10.48550/arxiv.2108.10545

openalex publication_date 2021/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let W be a real symplectic space and (G,G') an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let \widetildeG be the preimage of G in the metaplectic group \widetildeSp(W). Given an irreducible unitary representation Π of \widetildeG that occurs in the restriction of the Weil representation to \widetildeG, let Θ_Π denote its character. We prove that, for the embedding T of \widetildeSp(W) in the space of tempered distributions on W given by the Weil representation, the distribution T(\checkΘ_Π) has an asymptotic limit. This limit is an orbital integral over a nilpotent orbit \mathcal O_m⊆ W. The closure of the image of \mathcal O_m in \mathfrakg' under the moment map is the wave front set of Π', the representation of \widetildeG' dual to Π.

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