Appears in collection : Symmetry in Geometry and Analysis

Let $(G, G')$ be a reductive dual pair in $\mathrm{Sp}(W)$ with $\mathrm{rank} G \leq \mathrm{rank} G'$. The image of the Casimir element of the universal enveloping algebra of $G$ under the Weil representation $\omega$ is usually called the Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of $\omega$. We compute the resonances of this operator for the smallest split orthosymplectic dual pairs. The resonance representations turn out to be $GG'$-modules in Howe’s correspondence. We determine them explicitly.

This is joint work with Roberto Bramati (Ghent University) and Tomasz Przebinda (University of Oklahoma).