The density Hypothesis for irreducible lattices in $\mathrm{PSL}_2(\mathbb{R})^d$

Abstract

We prove the density hypothesis for congruence subgroups of an irreducible uniform lattice in $\mathrm{PSL}_2(\mathbb{R})^d$, extending previous results on the spherical density hypothesis to bound multiplicities of non-tempered non-spherical representations. Our bounds are uniform in the level as well as the spectral parameters.