Periods of automorphic forms over spherical subgroups tend to: (1) distinguish images of functorial lifts and (2) give information about L-functions.
This raises the following questions, given a spherical variety X=H\G: Locally, which irreducible representations admit a non-zero H-invariant functional or, equivalently, appear in the space of functions on X? Globally, can the period over H of an automorphic form on G be related to some L-value?
The conjectural answer involves a "dual group" associated to X and can be seen as a generalization of part of the Langlands conjectures for the case X=a group under left and right multiplication by itself. I will describe the dual group and discuss evidence suggesting that the relative trace formula of Jacquet is the correct framework for a more precise formulation of the conjectures.