Symmetry breaking operators for dual pairs with one member compact
M. McKee, A. Pasquale, T. Przebinda
TL;DR
This work analyzes symmetry breaking operators arising from Howe duality when one member of a reductive dual pair (G,G′) is compact. By encoding the unique symmetry breaking operator as an intertwining distribution f_{Π⊗Π′} and computing its Weyl symbol, the authors connect Howe’s correspondence to explicit Weyl calculus and orbital integrals on the symplectic space W. They derive two central, explicit formulas for f_{Π⊗Π′} in the regimes l < l′ and l ≥ l′, ascertain weight conditions for Π to occur in the correspondence, and obtain the wavefront set of the dual Π′ via elementary orbital techniques. The results also yield non-differential nature of the symmetry breaking operators and provide concrete calculations in important examples such as (O_2, Sp_{2l′}) and (U_l, U_{p,q}). Overall, the paper links explicit orbital data on W to representation-theoretic branching in Howe dual pairs, enriching both the theory of symmetry breaking and the practical determination of Howe correspondence constituents.
Abstract
We consider a dual pair $(G, G')$, in the sense of Howe, with G compact acting on $L^2(\mathbb{R}^n)$, for an appropriate $n$, via the Weil representation $ω$. Let $\tilde{\mathrm{G}}$ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation $Π$ of $\tilde{\mathrm{G}}$, let $Π'$ be the corresponding irreducible unitary representation of $\tilde{\mathrm{G}'}$ in the Howe duality. The orthogonal projection onto $L^2(\mathbb{R}^n)_Π$, the $Π$-isotypic component, is the essentially unique symmetry breaking operator in $\mathrm{Hom}_{\tilde{\mathrm{G}}\tilde{\mathrm{G}'}}(\mathcal{H}_ω^{\infty}, \mathcal{H}_Π^{\infty}\otimes \mathcal{H}_{Π'}^{\infty})$. 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 $\tilde{\mathrm{G}}$ occurring in Howe's correspondence when the rank of $\tilde{\mathrm{G}}$ is strictly bigger than the rank of $\tilde{\mathrm{G'}}$. They also allow us to compute the wavefront set of $Π'$ by elementary means.