Recent Developments in analysis on Spherical spaces

I will overview recent developments in harmonic analysis on spherical spaces. This class of spaces includes the class of symmetric spaces and recently became relevant for variations on the Langlands program
due to its relation to periods of automorphic forms.

I will introduce some basic geometric properties of these spaces and then focus on two issues: the decay of generalized matrix coefficients on real spherical spaces and the regularity of generalized (spherical) characters.

After reviewing the necessary background, we will discuss some of our results and elaborate on few applications of these results to problems originating in arithmetic. In particular we will discuss some new results on the problem of counting lattice points in the realm of real spherical spaces.

The main results include quantitative generalizations of Howe-Moore phenomena in the real case and a qualitative generalizations of Howe/Harish-Chandra character expansions in the p-adic case. Our techniques relies on systematic usage of the action of Bernstein center in the p-adic case and the theory of ODE in the real case (using the z-finite action of the center of the universal enveloping algebra).

The lecture is based on recent works with various collaborators (A. Aizenbud, D. Gourevtich, B. Kroetz, F. Knop, H. Schlichtkrull).