亚辛群作用于辛旋量值形式的Howe型对偶
Howe type duality for the metaplectic group acting on symplectic spinor valued forms
摘要：
设$\mathbb{S}$表示复辛李代数$\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},ω).$上的振荡模考虑值在振荡模中的外形式的$\mathfrak{g}$ -模$\mathbb{W}=\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$.证明了结合代数$\hbox{End}_{\mathfrak{g}}(\mathbb{W})$是由正交辛李超代数$\mathfrak{osp}(1|2)$的某种表示的象和两个特殊的投影算子生成的.将空间$\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$关于$\mathfrak{g}$和$\mathfrak{osp}(1|2).$的联合作用进行分解，建立了$\mathfrak{sp}(\mathbb{V}^{\mathbb{C}},ω)$作用于$\mathbb{W}.$的Howe型对偶
英文摘要：
Let $\mathbb{S}$ denote the oscillatory module over the complex symplectic Lie algebra $\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},ω).$ Consider the $\mathfrak{g}$-module $\mathbb{W}=\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$ of exterior forms with values in the oscillatory module. We prove that the associative algebra $\hbox{End}_{\mathfrak{g}}(\mathbb{W})$ is generated by the image of a certain representation of the ortho-symplectic Lie super algebra $\mathfrak{osp}(1|2)$ and two distinguished projection operators. The space $\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$ is decomposed with respect to the joint action of $\mathfrak{g}$ and $\mathfrak{osp}(1|2).$ This establishes a Howe type duality for $\mathfrak{sp}(\mathbb{V}^{\mathbb{C}},ω)$ acting on $\mathbb{W}.$
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