Annihilator varieties of distinguished modules of reductive Lie algebras

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