Whittaker supports for representations of reductive groups

Let $F$ be either $mathbb{R}$ or a finite extension of $mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=mathbb{R}$). For each nilpotent orbit $mathcal{O}$ we consider a certain Whittaker quotient $pi_{mathcal{O}}$ of $pi$. We define the Whittaker support WS$(pi)$ to be the set of maximal $mathcal{O}$ among those for which $pi_{mathcal{O}} eq 0$. In this paper we prove that all $mathcal{O}inmathrm{WS}(pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $pi$ is quasi-cuspidal then we show that all $mathcal{O}inmathrm{WS}(pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].