The resonances of the Capelli operators for small split orthosymplectic dual pairs
Résumé
Let (G,G') be a reductive dual pair in Sp(W) with rank G ≤ rank G' and G' semisimple. The image of the Casimir element of the universal enveloping algebra of G' under the Weil representation ω is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of ω. 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.
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