Dual groups of spherical varieties and an extension of the Langlands conjectures

Spherical varieties form a wide class of interesting (almost) homogeneous
spaces for reductive groups, which includes symmetric spaces, flag
varieties and others. If $X=H\backslash G$ is such a variety, the problem of
distinction asks when does an irreducible representation $\pi$ of $G(k)$, where 
$k$ is a local field, appear in the space of functions on $X$; and globally, 
for which automorphic representations of $G$ is the period integral over an 
orbit of $H$  non-zero. \\

I will explain how one attaches a dual group to a spherical variety
(following Gaitsgory and Nadler, and later work of Venkatesh and myself,
all based on results of Brion, Knop and others). And how this dual group
answers (conjecturally) some of the above questions, in a way that
generalizes some of the Langlands conjectures (which correspond to the
spherical variety $X=H, G=H \times H$).