Title: Bounds on multiplicities of spherical spaces over finite fields
Abstract:

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number M such that the multiplicity dim Hom(ρ,C[X(F)]) is bounded by M, for any finite field F and any irreducible representation ρ of G(F). 

We give an explicit bound for M.  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.