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Pré-Publication, Document De Travail Année : 2023

Symmetry breaking operators for dual pairs with one member compact

Angela Pasquale

Résumé

We consider a dual pair (G, G'), in the sense of Howe, with G compact acting on L^2(R^n), for an appropriate n, via the Weil representation $\omega$. Let \wt\G be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation \Pi of \wt\G, let \Pi' be the corresponding irreducible unitary representation of \wt\G' in the Howe duality. The orthogonal projection onto L^2(\R^n)_\Pi, the \Pi-isotypic component is, up to a constant multiple, the unique symmetry breaking operator Hom_{\wt\G \wt\G'}(\Hc_\omega^\infty, \Hc_\Pi^\infty\otimes \Hc_{\Pi'}^\infty). We study this operator by computing its Weyl symbol. Our results allow us to compute the wavefront set of \Pi' by elementary means.
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Dates et versions

hal-03293407 , version 1 (21-07-2021)
hal-03293407 , version 2 (12-02-2023)

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M Mckee, Angela Pasquale, T Przebinda. Symmetry breaking operators for dual pairs with one member compact. 2023. ⟨hal-03293407v2⟩
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