The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs
Journal of Lie Theory, Volume 33 (2023) no. 1, pp. 93-132
Let $(\mathrm{G},\mathrm{G})$ be a reductive dual pair in ${\rm Sp}(\mathsf{W})$ with ${\rm rank}\, \mathrm{G} \leq {\rm rank}\, \mathrm{G}$ and $\mathrm{G}'$ semisimple. The image of the Casimir element of the universal enveloping algebra of $\mathrm{G}'$ 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 $\mathrm{G}\mathrm{G}$-modules in Howe's correspondence. We determine them explicitly.
@article{JOLT_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},
doi = {10.5802/jolt.1276},
zbl = {1526.43008},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1276/}
}
TY - JOUR AU - R. Bramati AU - A. Pasquale AU - T. Przebinda TI - The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs JO - Journal of Lie Theory PY - 2023 SP - 93 EP - 132 VL - 33 IS - 1 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1276/ DO - 10.5802/jolt.1276 LA - en ID - JOLT_2023_33_1_a4 ER -
%0 Journal Article %A R. Bramati %A A. Pasquale %A T. Przebinda %T The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs %J Journal of Lie Theory %D 2023 %P 93-132 %V 33 %N 1 %U https://jolt.centre-mersenne.org/articles/10.5802/jolt.1276/ %R 10.5802/jolt.1276 %G en %F JOLT_2023_33_1_a4
R. Bramati; A. Pasquale; T. Przebinda. The Resonances of the Capelli Operators for Small Split Orthosymplectic Dual Pairs. Journal of Lie Theory, Volume 33 (2023) no. 1, pp. 93-132. doi: 10.5802/jolt.1276
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