论文标题
Schwartz的广义Whittaker在G空间上功能
Generalized Whittaker quotients of Schwartz functions on G-spaces
论文作者
论文摘要
让$ g $为当地字段$ f $的特征零（Archimedean）的还原组。令$ x $为$ g $ - 空间。在本文中，我们研究了Schwartz在$ x $上功能的广义Whittaker商的存在，被认为是$ G $的代表。我们表明，双重空间与谎言代数的一组nilpotent元素集，以使相应的广义惠特克商不会消失，其中包含矩映射图像的nilpotent部分，并且位于该图像的闭合。这概括了Prasad和Sakellaridis的最新结果。 将我们的定理应用于对称对$（g，h）$，我们表明存在一个无限维$ h $ distance的表示为$ g $的表示，并且只有当时与对应于该配对$ $ $（g，h）$相对应的真正还原群是非处理的。对于准切片$ g $，我们还扩展到阿基米德案例的prasad定理，指出存在$ g $的通用$ h $ distandistation $ g $的代表，并且仅当与对应于该配对$ $ $（g，h）$相对应的真实还原群体是quasi-split。 在非Archimedean情况下，我们的结果在杰出表示的波浪集集中还具有相当锐利的界限。 本文中的结果可用于在Ash-Ginzburg-Rallis证明的自动形式时期恢复许多消失的结果。这是从我们的推论H加上对[GGS21]中证明的Cuspidal Automorthic表示的惠特克支持的限制时的限制。
Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs $(G,H)$ we show that there exists an infinite-dimensional $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is non-compact. For quasi-split $G$ we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is quasi-split. In the non-Archimedean case our result also gives rather sharp bounds on the wave-front sets of distinguished representations. The results in the present paper can be used to recover many of the vanishing results on periods of automorphic forms proved by Ash-Ginzburg-Rallis. This follows from our Corollary H when combined with the restrictions on the Whittaker support of cuspidal automorphic representations proven in [GGS21].
