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