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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]

We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $${{G}}$$ defined over $$\mathbb {R}$$ , the space of $${{G(\mathbb {R})}}$$ -equivariant distributions on the manifold of real points of any algebraic $${{G}}$$ -manifold $${{X}}$$ is finite-dimensional if $${{G}}$$ has finitely many orbits on $${{X}}$$ . [ABSTRACT FROM AUTHOR]