Whittaker supports for representations of reductive groups

Let F be either R or a finite extension of Qp, and let G be a finite central extension of the group of F-points of a reductive group defined over F. Also let π be a smooth representation of G (Fréchet of moderate growth if F = R). For each nilpotent orbit O we consider a certain Whittaker quotient π_O of π. We define the Whittaker support WS(π) to be the set of maximal O among those for which π_O 6= 0. In this paper we prove that all O ∈ WS(π) are quasi-admissible nilpotent orbits, generalizing results of Mœglin and Jiang–Liu–Savin. If F is p-adic and π is quasicuspidal then we show that all O ∈ WS(π) 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 results of Ginzburg–Rallis–Soudry, Shen, and Cai, and confirming some conjectures of Ginzburg. Our methods are a synergy of the methods of the above-mentioned authors, and of our own earlier work.