Bounds on multiplicities of symmetric pairs of finite groups

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 ${\operatorname {dim}} \rho ^{\Gamma ^{\theta }}$ in terms of the smallest dimension of a faithful $\mathbb {F}_p$ -representation of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$ , where p is any odd prime and $\operatorname {\mathrm {Rad}}_p(\Gamma )$ 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.