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
Roberto.Bramati@UGent.be
Angela Pasquale
Université de Lorraine, Institut E. Cartan, Metz, France
angela.pasquale@univ-lorraine.fr
Tomasz Przebinda
Department of Mathematics, University of Oklahoma, Norman, U.S.A.
tprzebinda@ou.edu
[Abstract-pdf]
\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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