A residue map and a Poisson kernel for $GL_3$

Abstract: In the classical theory of modular and automorphic forms it has proven to be very useful to realize spaces of such forms in more combinatorial or algebraic ways. A famous instance of such a realization is the relationship between classical modular forms and modular symbols. In this talk, I will discuss a non-archimedean analogue of this construction, namely the relationship between certain holomorphic discrete series representations on the Drinfeld period domain and spaces of harmonic cocycles on the Bruhat-Tits building for the group $GL_3$ over a non-archimedean local field of any characteristic. The main novelty is that we allow non-trivial coefficients in a situation beyond the well-known theory for $GL_2$, which extends works of Schneider and Teitelbaum. I will explain how to construct a residue map and a Poisson kernel in this situation. Moreover, I will explain how the existence of the relevant boundary distributions follows from a conjectural non-criticality statement for certain (generalized) automorphic forms.

number theory

Audience: researchers in the discipline

Boston University Number Theory Seminar

Series comments: The seminar will now meet in CDS 365 (in the new building!). Tea begins at 3:30 in the same room.