A Conjecture of Sakellaridis-Venkatesh on the Unitary Spectrum of Spherical Varieties

In a recent preprint, Sakellaridis and Venkatesh considered the spectral decomposition of the space $L^2(X)$, where $X = H\G$ is a spherical variety and $G$ is a real or $p$-adic group, and stated a conjecture describing this decomposition in terms of a dual group $\check{G}_X$ associated to $X$. The main purpose of this paper is to verify the above conjecture in many cases when $X$, has low rank. In particular, we demonstrate this conjecture for many cases when $X$ has rank 1, and also some cases when $X$ has rank 2 or 3.