Bounds on multiplicities of symmetric pairs of finite groups

TL;DR

This paper establishes bounds on the multiplicities of irreducible representations fixed by involutions in finite groups, with implications for group schemes over integers and their representations over p-adic integers.

Contribution

It introduces bounds on the dimensions of fixed-point spaces of irreducible representations under involutions, linking them to minimal faithful representations over finite fields.

Findings

Bound on $ ext{dim} ho^{ ext{fixed subgroup}}$ in terms of faithful representations.

Uniform bounds on multiplicities in p-adic group representations.

Application to group schemes over $ ext{Z}$ and their representations over $ ext{Z}_p$.

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 $p$.