Sarnak–Xue density hypothesis

Let $G\leq GL_n$ be a connected semisimple linear algebraic group over $\mathbb{Q}$ with non-compact $G_\infty=G(\mathbb{R})$. For $q\in\mathbb{N}$, let $\Gamma(q)$ be the level-$q$ congruence subgroup, let $\Pi^{\mathrm{unit}}(G_\infty)$ denote the irreducible unitary representations of $G_\infty$, let $r(\pi)\in[2,\infty]$ be the rate of decay of matrix coefficients, and let $m(\pi;q)$ be the multiplicity of $\pi$ in $L^2(\Gamma(q)\backslash G_\infty)$. Sarnak–Xue density hypothesis. For any $\pi\in\Pi^{\mathrm{unit}}(G_\infty)$,

$$m(\pi;q)\ll\operatorname{vol}(\Gamma(q)\backslash G_\infty)^{\frac{2}{r(\pi)}}.$$

This is the classical density hypothesis relating multiplicities in congruence quotients to the decay of matrix coefficients; its status is not specified in the source.