  In this paper, we will determine the factorization of global period attached
to the spherical variety $X:=\mathrm{SO}(n-1)\backslash \mathrm{SO}(n)$, which
is a special case of the Sakellaridis-Venkatesh conjecture. The main idea is to
build a connection between the periods of $X$ and the periods of the Whittaker
case $N,\psi\backslash \mathrm{SL}_2$ (n even) or $N,\psi\backslash
\mathrm{Mp}_2$ (n odd) using the tool of theta correspondence. In the local
setting, we determine the Plancherel formula of $L^2(X)$, the relative
character identities and give an explicite integral formula of transfer between
$X$-side and Whittaker-side which coincides with the theory of transfer
developed by Sakellaridis.
