The resonances of the Capelli operators for small split orthosymplectic dual pairs

TL;DR

This paper computes the resonances of Capelli operators associated with small split orthosymplectic dual pairs within the Weil representation, explicitly identifying the resulting resonance representations in Howe's correspondence.

Contribution

It explicitly determines the resonance representations of Capelli operators for small split orthosymplectic dual pairs, linking them to Howe's correspondence.

Findings

Resonance representations are identified as $GG'$-modules.

Explicit computation of resonances for specific dual pairs.

Resonance representations are characterized within Howe's correspondence.

Abstract

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