Sarnak–Xue multiplicity conjecture for unitary representations

Let $G$) be a semisimple Lie group, let $\Gamma(\mathfrak{p}^n)\subset G$ be a congruence tower of cocompact arithmetic lattices, and let $\pi$ be a unitary representation of $G$. Define

$$p(\pi)=\inf\{p\geq 2\mid \text{the $K$-finite matrix coefficients of $\pi$ are in $L^p(G)$}\}.$$

Sarnak–Xue conjecture. The multiplicity of $\pi$ satisfies

$$m(\pi,\mathfrak{p}^n)\ll_{\epsilon}[\Gamma(\mathfrak{p}^n):\Gamma(1)]^{\frac{2}{p(\pi)}+\epsilon}.$$

This conjecture predicts upper bounds interpolating between the proportional-to-index growth of discrete-series multiplicities and the constant multiplicity of the trivial representation. It concerns the asymptotic growth of multiplicities in congruence towers and, in particular, has applications to the growth of cohomology of cocompact arithmetic subgroups.