  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].
