Local Theta Lifting of Generalized Whittaker Models Associated to Nilpotent Orbits

Abstract

Let (G, G̃) be a reductive dual pair over a local field (Formula presented.) of characteristic 0, and denote by V and (Ṽ the standard modules of G and G̃, respectively. Consider the set Max Hom (V, Ṽ) of full rank elements in Hom(V, (Ṽ), and the nilpotent orbit correspondence (Formula presented.) and (Formula presented.) induced by elements of Max Hom (V, Ṽ) via the moment maps. Let (Formula presented.) be a smooth irreducible representation of G. We show that there is a correspondence of the generalized Whittaker models of π of type (Formula presented.) and of Θ (π) of type (Formula presented.), where Θ (π) is the full theta lift of π. When (G, G̃) is in the stable range with G the smaller member, every nilpotent orbit (Formula presented.) is in the image of the moment map from Max Hom (V, Ṽ). In this case, and for (Formula presented.) non-Archimedean, the result has been previously obtained by Mglin in a different approach. © 2014 Springer Basel.