The main goal of this talk is to state a conjectural formula for the compactly supported cohomology of  \Gamma \ G / K  when G is a semisimple real group with maximal compact K, and \Gamma is a congruence subgroup.  The formula gives a (conjectural!) expression for this cohomology in terms of objects appearing on the spectral/Galois side of various categorical local Langlands conjectures, and generalizes to the non-Shimura variety context a formula already proposed by myself and Xinwen Zhu in the case of Shimura varieties. Such a formula is in fact a consequence of very general categorical Langland principles, as proposed by Peter Scholze. However, going from these general principles to an explicit conjecture involves implementing a version of the local categorical correspondence at infinity.

In this presentation I will propose a concrete conjecture (under the additional assumption that G is an adjoint group), which connects this problem in a surprising way (surprising to me, at least) to Vogan's theory of small K-types (actually K-hat types, since we will apply this theory on the Langlands dual side). 

If time allows, I also hope to say something about the relationship b/w this explicit proposal and the general principles already mentioned. This is joint work with Dougal Davis and Kari Vilonen