论文标题
Capelli操作员的共鸣，用于小型分裂的正交双对
The resonances of the Capelli operators for small split orthosymplectic dual pairs
论文作者
论文摘要
令$（g，g'）$为$ sp（w）$的还原双对，而排名$ g \ leq $ stark $ g'$和$ g'$ semisimple。在Weil表示$ω$下，通用包络代数的Casimir元素的图像是Capelli操作员。它是一个遗传操作员，作用于$ω$的表示空间的平滑向量。我们计算该操作员翻译的自然倍数的共振，用于小型分裂的正交双对。事实证明，相应的共振表示为$ gg'$ - 模块中的模块。我们明确确定它们。
Let $(G,G')$ be a reductive dual pair in $Sp(W)$ with rank $G\leq$ 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.
