Symmetry Breaking Operators for Dual Pairs with One Member Compact
Vol. 32 (2025), No. 3, Page 257–400.
McKee, M.; Pasquale, A.; Przebinda, T.
Symmetry Breaking Operators for Dual Pairs with One Member Compact
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Abstract:
We consider a dual pair ($\rm{G,G'}$), in the sense of Howe, with $\rm{G}$ compact acting on $\rm{L}^2(\mathbb{R}^n)$, for an appropriate $n$, via the Weil representation $\omega$. Let $\rm\tilde{G}$ be the preimage of $\rm{G}$ in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\rm\tilde{G}$, let $\Pi'$ be the corresponding irreducible unitary representation of $\rm\tilde{G}'$ in Howe's correspondence. The orthogonal projection onto the $\Pi$-isotypic component $\rm{L}^2(\mathbb{R}^n)_\Pi$ is, up to a constant multiple, the unique symmetry breaking operator in $\rm{Hom}_{\rm\tilde{G}\rm\tilde{G}'}(\mathcal{H}_\omega^\infty, \mathcal{H}_\Pi^\infty\otimes \mathcal{H}_{\Pi'}^\infty)$. We study this operator by computing its Weyl symbol. Our results allow us to recover the known list of highest weights of irreducible representations of $\rm\tilde{G}$ occurring in Howe's correspondence when the rank of $\rm{G}$ is strictly bigger than the rank of $\rm{G}'$. They also allow us to compute the wavefront set of $\Pi'$ by elementary means.
Keywords: Reductive dual pairs, Howe duality, symmetry breaking operators, Weyl calculus, Lie superalgebras.
Received: 2021-07-20
