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We provide a micro-local necessary condition for distinction of admissiblerepresentations of real reductive groups in the context of spherical pairs.  Let $\bf G$ be a complex algebraic reductive group, and $\bf H\subset G$ be aspherical algebraic subgroup. Let $\mathfrak{g},\mathfrak{h}$ denote the Liealgebras of $\bf G$ and $\bf H$, and let $\mathfrak{h}^{\bot}$ denote theannihilator of $\mathfrak{h}$ in $\mathfrak{g}^*$. A $\mathfrak{g}$-module iscalled $\mathfrak{h}$-distinguished if it admits a non-zero$\mathfrak{h}$-invariant functional. We show that the maximal $\bf G$-orbit inthe annihilator variety of any irreducible $\mathfrak{h}$-distinguished$\mathfrak{g}$-module intersects $\mathfrak{h}^{\bot}$. This generalizes aresult of Vogan.  We apply this to Casselman-Wallach representations of real reductive groupsto obtain information on branching problems, translation functors and Jacquetmodules. Further, we prove in many cases that as suggested by Prasad, if $H$ isa symmetric subgroup of a real reductive group $G$, the existence of a tempered$H$-distinguished representation of $G$ implies the existence of a generic$H$-distinguished representation of $G$.  Many models studied in the theory of automorphic forms involve an additivecharacter on the unipotent radical of $\bf H$, and we devised a twisted versionof our theorem that yields necessary conditions for the existence of thosemixed models. Our method of proof here is inspired by the theory of W-algebras.As an application we derive necessary conditions for the existence ofRankin-Selberg, Bessel, Klyachko and Shalika models. Our results are compatiblewith the recent Gan-Gross-Prasad conjectures for non-generic representations.  We also prove more general results that ease the sphericity assumption on thesubgroup, and apply them to local theta correspondence in type II and todegenerate Whittaker models.

Annihilator varieties of distinguished modules of reductive Lie algebras