Local theta lifting of generalized Whittaker models associated to nilpotent orbits

TL;DR

This paper establishes a correspondence between generalized Whittaker models of representations in reductive dual pairs over local fields, linking nilpotent orbit types via local theta lifts, extending known results in the stable range.

Contribution

It proves a new correspondence of generalized Whittaker models under local theta lifting for reductive dual pairs, generalizing prior work in the stable range.

Findings

Established a correspondence between Whittaker models of type $ ext{O}$ and $ ext{Θ}( ext{O})$.

Proved the result for non-Archimedean fields in the stable range, extending previous work.

Connected nilpotent orbit correspondence with theta lifts in the context of reductive dual pairs.

Abstract

Let $(G,\tilde{G})$ be a reductive dual pair over a local field ${\Fontauri k}$ of characteristic 0, and denote by $V$ and $\tilde{V}$ the standard modules of $G$ and $\tilde{G}$, respectively. Consider the set $Max Hom(V,\tilde{V})$ of full rank elements in $Hom(V,\tilde{V})$, and the nilpotent orbit correspondence $\mathcal{O} \subset \mathfrak{g}$ and $\Theta (\mathcal{O})\subset \tilde{\mathfrak{g}}$ induced by elements of $Max Hom(V,\tilde{V})$ via the moment maps. Let $(\pi,\mathscr{V})$ be a smooth irreducible representation of $G$. We show that there is a correspondence of the generalized Whittaker models of $\pi $ of type $\mathcal{O}$ and of $\Theta (\pi)$ of type $\Theta (\mathcal{O})$, where $\Theta (\pi)$ is the full theta lift of $\pi $. 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 $Max Hom (V,\tilde{V})$. In this case, and for ${\Fontauri k}$ non-Archimedean, the result has been previously obtained by M\oe{}glin in a different approach.