G-空间上Schwartz函数的广义Whittaker导数
Generalized Whittaker quotients of Schwartz functions on G-spaces
摘要：
设$G$是特征为零的局部域$F$上的约化群，无论是否为阿基米德。设$X$是一个$G$-空间。本文研究了$X$上Schwartz函数空间的广义Whittaker子空间的存在性，它被认为是$G$的一种表示.证明了使得相应的广义Whittaker商不为零的李代数的对偶空间的幂零元的集合包含矩映射的象的幂零部分，并且存在于该象的闭包中.这一结果推广了Prasad和Sakellarlane最近的结果。将我们的定理应用于对称对$(G,H)$，我们证明了存在$G$的无限维$H$-区别表示当且仅当对应于对称对$(G,H)$的实约化群是非紧的.对于拟分裂$G$，我们还扩展到阿基米德的情况下，普拉萨德定理指出，存在一个通用的$H$-区别表示$G$当且仅当真正的约化群对应的对$(G,H)$是拟分裂。在非阿基米德的情况下，我们的结果也对不同表示的波前集给出了相当尖锐的界限。本文的结果可以用来恢复Ash-Ginzburg-Rallis证明的关于自守形式周期的许多消失结果。这是从我们的推论H，结合在[GGS21]中证明的尖点自守表示的Whittaker支持的限制。
英文摘要：
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].
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