Locally Symmetric Spaces Seminar

I will talk about some global questions on the concentration properties of automorphic forms that seem to be a natural environment in which to examine the interplay between the geometry and spectrum of spherical varieties, L-functions, and functoriality. These questions are basically concerned with the microlocal behavior of automorphic forms, although (to my knowledge) they have not yet been expressed in that language: for example, we work with automorphic forms on the locally symmetric space itself, rather than with their microlocal lifts to the tangent bundle. One of the goals is to develop good conjectures, supported by all existing examples, possibly based on the emerging framework of the moment map, as in this working group. As a starting point, I will give an exposition of Sections 2 and 3 in my joint paper with Simon Marshall, "Lower bounds on Maass forms on semisimple groups". These sections examined the question of large sup norms (concentration at points) in relation with the conjectures of Sakallaridis-Venkatesh on the spectrum of spherical varieties. If time (and energy) permits, I then hope to adapt the viewpoint taken in that article to ongoing work with Marshall on concentration behavior along geodesics.