  Given a spherical variety X for a group G over a non-archimedean local field
k, the Plancherel decomposition for L^2(X) should be related to "distinguished"
Arthur parameters into a dual group closely related to that defined by
Gaitsgory and Nadler. Motivated by this, we develop, under some assumptions on
the spherical variety, a Plancherel formula for L^2(X) up to discrete (modulo
center) spectra of its "boundary degenerations", certain G-varieties with more
symmetries which model X at infinity. Along the way, we discuss the asymptotic
theory of subrepresentations of C^{infty}(X), and establish conjectures of
Ichino-Ikeda and Lapid-Mao. We finally discuss global analogues of our local
conjectures, concerning the period integrals of automorphic forms over
spherical subgroups.
