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Journal of Lie Theory 33 (2023), No. 1, 093--132
Copyright Heldermann Verlag 2023

The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs

Roberto Bramati
Department of Mathematics, Ghent University, Ghent, Belgium

Angela Pasquale
Université de Lorraine, Institut E. Cartan, Metz, France

Tomasz Przebinda
Department of Mathematics, University of Oklahoma, Norman, U.S.A.


\def\G{\mathrm{G}} \def\Wv{\mathsf{W}} Let $(\G,\G’)$ be a reductive dual pair in ${\rm Sp}(\Wv)$ with ${\rm rank}\, \G \leq {\rm rank}\, \G’$ 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 $\G\G’$-modules in Howe's correspondence. We determine them explicitly.

Keywords: Resonances, Capelli operators, Howe's correspondence.

MSC: 43A85, 58J50, 22E30.

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