Bounds on multiplicities of symmetric pairs of finite groups

Abstract

Let $Γ$ be a finite group, let $θ$ be an involution of $Γ$, and let $ρ$ be an irreducible complex representation of $Γ$. We bound $\dim ρ^{Γ^θ}$ in terms of the smallest dimension of a faithful $\mathbb{F}_p$-representation of $Γ/Rad_p(Γ)$, where $p$ is any odd prime and $Rad_p(Γ)$ is the maximal normal $p$-subgroup of $Γ$. This implies, in particular, that if $\mathbf{G}$ is a group scheme over $\mathbb{Z}$ and $θ$ is an involution of $\mathbf{G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf{G}(\mathbb{Z}_p)/ \mathbf{G} ^θ(\mathbb{Z}_p) \right)$ is bounded, uniformly in $p$.