Geodesic
The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs
Journal of Lie theory, Tome 33 (2023) no. 1, pp. 93-132
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\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.
Classification : 43A85, 58J50, 22E30
Mots-clés : Resonances, Capelli operators, Howe's correspondence 
@article{JLT_2023_33_1_JLT_2023_33_1_a4,
     author = {R. Bramati and A. Pasquale and T. Przebinda},
     title = {The {Resonances} of the {Capelli} {Operators} for {Small} {Split} {Orthosymplectic} {Dual} {Pairs}},
     journal = {Journal of Lie theory},
     pages = {93--132},
     year = {2023},
     volume = {33},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2023_33_1_JLT_2023_33_1_a4/}
}
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R. Bramati; A. Pasquale; T. Przebinda. The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs. Journal of Lie theory, Tome 33 (2023) no. 1, pp. 93-132. http://geodesic.mathdoc.fr/item/JLT_2023_33_1_JLT_2023_33_1_a4/