arXiv subjects

M. McKee

Publications and source records attributed to M. McKee.

5 recordsLinked to original sources

arXiv2023

Symmetry breaking operator for the reductive dual pair $(\mathrm{U}_l,\mathrm{U}_{l'})$

We consider the dual pair $(G,G')=(\mathrm{U}_l,\mathrm{U}_{l'})$ in the symplectic group $\mathrm{Sp}_{2ll'}(\mathbb{R})$. Fix a Weil representation of the metaplectic group $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})$. Let $\tilde{G\,}$ and $\tilde{G'}$ be the preimages of $G$ and $G'$ under the metaplectic cover $\tilde{\mathrm{Sp}}_{2ll'}(\mathbb{R})\to \mathrm{Sp}_{2ll'}(\mathbb{R})$, and let $\Pi\otimes\Pi'$ be a genuine irreducible representation of $\tilde{G\,}\times\tilde{G'}$. We study the Weyl symbol $f_{\Pi\otimes\Pi'}$ of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with $\Pi\otimes\Pi'$. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its $\Pi$-isotypic component and also with the orthogonal projection onto its $\Pi'$-isotypic component. Hence $f_{\Pi\otimes\Pi'}$ can be computed in two different ways, one using $\Pi$ and the other using $\Pi'$. By matching the results, we recover Weyl's theorem stating that $\Pi\otimes\Pi'$ occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations $\Pi\otimes\Pi'$ occurring in Howe's correspondence.

math.RT↗

arXiv2023

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

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 $\widetilde{\mathrm{G}}$ be the preimage of G in the metaplectic group $\widetilde{\mathrm{Sp}}(\mathrm{W})$. Given an irreducible unitary representation $Π$ of $\widetilde{\mathrm{G}}$ that occurs in the restriction of the Weil representation to $\widetilde{\mathrm{G}}$, let $Θ_Π$ denote its character. We prove that, for the embedding $T$ of $\widetilde{\mathrm{Sp}}(\mathrm{W})$ 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 $\mathcal O_m\subseteq \mathrm{W}$. The closure of the image of $\mathcal O_m$ in $\mathfrak{g}'$ under the moment map is the wave front set of $Π'$, the representation of $\widetilde{\mathrm{G}'}$ dual to $Π$.

math.RT↗

arXiv2021

Symmetry breaking operators for dual pairs with one member compact

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 $\omega$. Let $\tilde{\mathrm{G}}$ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\tilde{\mathrm{G}}$, let $\Pi'$ be the corresponding irreducible unitary representation of $\tilde{\mathrm{G}'}$ in the Howe duality. The orthogonal projection onto $L^2(\mathbb{R}^n)_\Pi$, the $\Pi$-isotypic component, is the essentially unique symmetry breaking operator in $\mathrm{Hom}_{\tilde{\mathrm{G}}\tilde{\mathrm{G}'}}(\mathcal{H}_\omega^{\infty}, \mathcal{H}_\Pi^{\infty}\otimes \mathcal{H}_{\Pi'}^{\infty})$. 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 $\tilde{\mathrm{G}}$ 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 $\Pi'$ by elementary means.

math.RT↗

arXiv2015

Weyl calculus and dual pairs

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. Let $\widetilde{G}$ be the preimage of $G$ in the metaplectic group. Given a genuine irreducible unitary representation $Π$ of $\widetilde{G}$ we compute the Weyl symbol of orthogonal projection onto $L^2(\mathbb R^n)_Π$, the $Π$-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation $Π'$ of $\widetilde{G'}$ and to compute of the wave front set of $Π'$ by elementary means.

math.RT↗

arXiv2014

Semisimple orbital integrals on the symplectic space for a real reductive dual pair

We prove a Weyl Harish-Chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space $W$. As an intermediate step, we introduce a notion of a Cartan subspace and a notion of an almost semisimple element in the symplectic space $W$. We prove that the almost semisimple elements are dense in $W$. Finally, we provide estimates for the orbital integrals associated with the different Cartan subspaces in $W$.

math.RT↗