Almost multiplicity-one property of spherical varieties over finite fields

Let $H$ be a connected algebraic subgroup of a connected reductive group $G$ over a finite field $\mathbb F_q$ such that $G/H$ is a $G$-spherical variety, i.e., $G/H$ has an open dense $B$-orbit for each Borel subgroup $B$ of $G$.

We formulate, for the pair $(G,H)$, an almost multiplicity-one property.

Then we establish a criterion for this property in terms of the $B$-stabilizers on $G/H$.

In particular, we will see that this property is analogous to the strongly tempered condition in characteristic $0$.