arXiv 1302.37442013-02math.RT

与幂零轨道相关的广义Whittaker模型的局部theta提升

Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Raul Gomez, Chen-Bo Zhu

备注：: To appear in Geometric and Functional Analysis

摘要：

设$(G,\tilde{G})$是特征为0的局部域${\Fontauri k}$上的约化对偶对，用$V$和$\tilde{V}$分别表示$G$和$\tilde{G}$的标准模.考虑$Hom(V,\tilde{V})$中满秩元素的集合$Max Hom(V,\tilde{V})$，以及由$Max Hom(V,\tilde{V})$中的元素通过矩映射导出的幂零轨道对应$\mathcal{O} \subset \mathfrak{g}$和$Θ(\mathcal{O})\subset \tilde{\mathfrak{g}}$.设$(π,\mathscr{V})$是$G$的光滑不可约表示。我们表明，有一个对应的广义Whittaker模型的$π$的类型$\mathcal{O}$和$Θ(π)$的类型$Θ(\mathcal{O})$，其中$Θ(π)$是全θ电梯的$π$。当$(G,\tilde{G})$处于稳定区域内，$G$为小成员时，每个幂零轨道$\mathcal{O} \subset \mathfrak{g}$都在$Max Hom (V,\tilde{V})$的矩映射的像中.在这种情况下，对于非阿基米德的${\Fontauri k}$，结果已经由莫林用不同的方法得到。

英文摘要：

Let $(G,\tilde{G})$ be a reductive dual pair over a local field ${\Fontauri k}$ of characteristic 0, and denote by $V$ and $\tilde{V}$ the standard modules of $G$ and $\tilde{G}$, respectively. Consider the set $Max Hom(V,\tilde{V})$ of full rank elements in $Hom(V,\tilde{V})$, and the nilpotent orbit correspondence $\mathcal{O} \subset \mathfrak{g}$ and $Θ(\mathcal{O})\subset \tilde{\mathfrak{g}}$ induced by elements of $Max Hom(V,\tilde{V})$ via the moment maps. Let $(π,\mathscr{V})$ be a smooth irreducible representation of $G$. We show that there is a correspondence of the generalized Whittaker models of $π$ of type $\mathcal{O}$ and of $Θ(π)$ of type $Θ(\mathcal{O})$, where $Θ(π)$ is the full theta lift of $π$. When $(G,\tilde{G})$ is in the stable range with $G$ the smaller member, every nilpotent orbit $\mathcal{O} \subset \mathfrak{g}$ is in the image of the moment map from $Max Hom (V,\tilde{V})$. In this case, and for ${\Fontauri k}$ non-Archimedean, the result has been previously obtained by Mœglin in a different approach.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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