论文标题
杰出的还原谎言模块的歼灭者品种
Annihilator varieties of distinguished modules of reductive Lie algebras
论文作者
论文摘要
我们为在球形对的背景下为实际还原基团的可接受表示提供了微本地必要条件。 令$ \ bf g $为一个复杂的代数还原组，$ \ bf h \ subset g $是球形代数子组。令$ \ mathfrak {g}，\ mathfrak {h} $表示$ \ bf g $和$ \ bf h $的谎言代数，然后让$ \ mathfrak {h}^{\ bot} $代表$ \ \ \ \ \ \ \ \ \ \ \ \ \ \^$ nihihilator of $ \ mathfrak in $ \ \ \ nihihilator $ \ mathfrak {g} $ - module称为$ \ mathfrak {h} $ - 如果它允许非零$ \ mathfrak {h} $ - 不变函数。我们表明，任何不可约合$ \ mathfrak {h} $ - 区分$ \ mathfrak {g} $ - module Intersects $ \ mathfrak {h}^{h}^{\ bot} $。这概括了Vogan的结果。 我们将其应用于实际还原组的Casselman-Wallach表示，以获取有关分支问题，翻译函数和Jacquet模块的信息。此外，在许多情况下，正如Prasad所建议的，如果$ h $是真正的还原组$ g $的对称子组，则存在$ g $的恢复性$ h $ distance的代表，这意味着存在通用$ h $ h $ h $ distance的代表$ g $。 在自动形式理论中研究的许多模型都涉及$ \ bf h $的一能激进分子上的加性特征，我们设计了定理的扭曲版本，该版本为存在这些混合模型的存在带来了必要的条件。我们这里的证明方法是受W-Algebras理论的启发。作为一种应用，我们得出了Rankin-Selberg，Bessel，Klyachko和Shalika模型存在的必要条件。我们的结果与最新的Gan-gross-Prasad猜想兼容，用于非类别表示。 我们还证明了更一般的结果，可以简化子组上的球形假设，并将其应用于II型和退化Whittaker模型中的局部Theta对应关系。
We provide a micro-local necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let $\bf G$ be a complex algebraic reductive group, and $\bf H\subset G$ be a spherical algebraic subgroup. Let $\mathfrak{g},\mathfrak{h}$ denote the Lie algebras of $\bf G$ and $\bf H$, and let $\mathfrak{h}^{\bot}$ denote the annihilator of $\mathfrak{h}$ in $\mathfrak{g}^*$. A $\mathfrak{g}$-module is called $\mathfrak{h}$-distinguished if it admits a non-zero $\mathfrak{h}$-invariant functional. We show that the maximal $\bf G$-orbit in the annihilator variety of any irreducible $\mathfrak{h}$-distinguished $\mathfrak{g}$-module intersects $\mathfrak{h}^{\bot}$. This generalizes a result of Vogan. We apply this to Casselman-Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that as suggested by Prasad, if $H$ is a symmetric subgroup of a real reductive group $G$, the existence of a tempered $H$-distinguished representation of $G$ implies the existence of a generic $H$-distinguished representation of $G$. Many models studied in the theory of automorphic forms involve an additive character on the unipotent radical of $\bf H$, and we devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of W-algebras. As an application we derive necessary conditions for the existence of Rankin-Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan-Gross-Prasad conjectures for non-generic representations. We also prove more general results that ease the sphericity assumption on the subgroup, and apply them to local theta correspondence in type II and to degenerate Whittaker models.
