Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Let $(G,\tilde{G})$ be a reductive dual pair over a local field ${\Fontauri k}$ofcharacteristic0,anddenoteby$V$and$\tilde{V}$ the standard modules of $G$ and $\tilde{G}$, respectively. Consider the set $MaxHom(V,\tilde{V})$ of full rank elements in $Hom(V,\tilde{V})$, and the nilpotent orbit correspondence $\mathcal{O}\subset \mathfrak{g}$ and $\Theta ($\mathcal{O}$)\subset \tilde{\mathfrak{g}}$inducedbyelementsof$Max Hom(V,\tilde{V})$viathemomentmaps\mathrm{.}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 $\Theta (\pi)$oftype$\Theta ($\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 $MaxHom(V,\tilde{V})$. In this case, and for $\text{Fontauri}k$ non-Archimedean, the result has been previously obtained by M\oe{}glin in a different approach.