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 $\Gamma /Ra{d}_p(\Gamma )$, where $p$ is any odd prime and $Ra{d}_p(\Gamma )$ 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 $p$.

44 pages; Version 2 - minor corrections

References in corpus (4)

• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F
• Multiplicity one theorem for (GL(n+1,R),GL(n,R))
• Bounds on multiplicities of spherical spaces over finite fields -- the general case
• The Gelfand-Kazhdan criterion as a necessary and sufficient criterion