2016/10/02 by Gomez, Raul, Gourevitch, Dmitry, Sahi, Siddhartha
#17B08 #20G05 #20G20 #20G25 #20G30 #20G35 #22E27 #22E46 #22E50 #22E55 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1610.00284
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 (Frechet of moderate growth if F=ℝ). For each nilpotent orbit O we consider a certain Whittaker quotient πO of π. We define the Whittaker support WS(π) to be the set of maximal O among those for which πO≠ 0. In this paper we prove that all O\inWS(π) are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If F is p-adic and π is quasi-cuspidal then we show that all O\inWS(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 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].