Colloquium: Tuesday, November 1, 2 pm. Speaker: Eitan Sayag (Ben Gurion). Title: “Spherical functions, spherical distributions and their applications”.

Periods of automorphic forms provides an important tool to classify automorphic representations and arise time and time again in problems with arithmetic origin.

Motivated by that, I shall describe some of our investigation of local counterparts of these periods that involve representation theory and invariant harmonic analysis on p-adic and real spherical spaces. The results I plan to report on include:


* Quantitative generalizations of Howe-Moore phenomena regarding decay of generalized matrix coefficients in the real case.

* Qualitative generalizations of Howe/Harish-Chandra character expansions in the p-adic case.


I shall explain how to use the results on generalized matrix coefficients to obtain new results on counting lattice points in the realm of real Spherical spaces.

If time permits I will explain the role of Bernstein center in studying distributions in the p-adic case allowing tight control on the singularities of generalized characters on p-adic Spherical spaces.


My lecture will be based on joint works with B. Kr\”{o}tz, F. Knop and H. Schlichtkrull regarding decay of functions and on a joint work with Avraham Aizenbud and Dmitry Gourevitch regarding the regularity of certain distributions.