On the density hypothesis in higher rank

The non-tempered part of the $L^2$-spectrum of (finite volume) locally symmetric spaces plays an exceptional role in many applications. However, this part of the spectrum (besides the constant function) is very difficult to understand and is the object of some deep conjectures in the field of automorphic forms, such as the (generalized) Ramanujan conjecture. Sometimes good quantitative estimates on the size of the non-tempered spectrum are sufficient in practice. This leads to Sarnak's density hypothesis in higher rank.

In this talk we will discuss recent advances towards the density hypothesis for $\text{GL}_n$. In particular, we will focus on the level family defined by a Borel-type congruence subgroups.