Bounds on multiplicities of spherical spaces over finite fields

Let $G$ be a reductive group scheme of type $A$ acting on a spherical scheme $X$. We prove that there exists a number $C$ such that the multiplicity $\dim Hom(ρ,\mathbb{C}[X(F)])$ is bounded by $C$, for any finite field $F$ and any irreducible representation $ρ$ of $G(F)$. We give an explicit bound for $C$. We conjecture that this result is true for any reductive group scheme and when $F$ ranges (in addition) over all local fields of characteristic $0$.

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