Bounds on multiplicities of symmetric pairs of finite groups

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