Bound on multiplicities of symmetric pairs over p-adic fields

Abstract

We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group, the space, and the representation, making exact computations often difficult to carry out. This motivates the search for bounds depending only on structural invariants of the group and the field. More precisely, let $\mathbf{G}$ be a connected reductive group over a $p$-adic field $F$ of large residue characteristic (relative to the rank of $\mathbf{G}$), let $ρ$ be a smooth admissible irreducible representation of $G = \mathbf{G}(F)$ and let $θ$ be a rational involution with fixed-point subgroup $H=G^θ$. We show that the multiplicity \[ \dim \operatorname{Hom}_G\big(ρ, C^\infty(G/H)\big) \] is uniformly bounded. The bound depends only on the rank of $\mathbf{G}$ and the residue degree of $F$. Our approach combines Mackey theory with cohomological methods. As an application, we deduce a uniform bound on such multiplicities where the base field $F$ varies over all but a bounded number of local completions of a fixed number field.