## Graphs of Hecke operators

Roberto Alvarenga
IMPA
Abstract:
The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PHD thesis for
$PGL(2)$
. In this talk, we consider the generalization of this definition to
$GL(n)$
.
After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to explicitly calculate these graphs. We will conclude the talk with a schematic outline of how to perform such calculations.