Local theta lifting of generalized Whittaker models associated to nilpotent orbits.

Let $${(G,\tilde{G})}$$ be a reductive dual pair over a local field $${\mathfrak{k}}$$ of characteristic 0, and denote by V and $${\tilde{V}}$$ the standard modules of G and $${\tilde{G}}$$ , respectively. Consider the set $${{\rm Max} {\rm Hom} (V,\tilde{V})}$$ of full rank elements in $${{\rm Hom}(V,\tilde{V})}$$ , and the nilpotent orbit correspondence $${\mathcal{O} \subset \mathfrak{g}}$$ and $${\Theta (\mathcal{O})\subset \tilde{\mathfrak{g}}}$$ induced by elements of $${{\rm Max} {\rm Hom} (V,\tilde{V})}$$ via the moment maps. Let $${(\pi,\fancyscript{V})}$$ be a smooth irreducible representation of G. We show that there is a correspondence of the generalized Whittaker models of π of type $${\mathcal{O}}$$ and of Θ ( π) of type $${\Theta (\mathcal{O})}$$ , where Θ ( π) is the full theta lift of π. When $${(G,\tilde{G})}$$ is in the stable range with G the smaller member, every nilpotent orbit $${\mathcal{O} \subset \mathfrak{g}}$$ is in the image of the moment map from $${{\rm Max} {\rm Hom} (V,\tilde{V})}$$ . In this case, and for $${\mathfrak{k}}$$ non-Archimedean, the result has been previously obtained by Mœglin in a different approach. [ABSTRACT FROM AUTHOR]