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 C such that the multiplicity dim Hom (ρ , 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. Different aspects of this conjecture were studied in [3,11,6,7]. [ABSTRACT FROM AUTHOR]