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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.

Local theta lifting of generalized Whittaker models associated to nilpotent orbits