On graphs of Hecke operators

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms. Let $X$ be a curve over $\mathbb{F}_q$, $F$ its function field and $\mathbb{A}$ the adele ring of $F$. In this paper we will exhibit the first properties for the graph of Hecke operators for $\mathrm{GL}_n(\mathbb{A}),$ for every $n \geq 1.$ This includes a description of the graph in terms of coherent sheaves on $X.$ We provide a numerical condition for two vertices to be connected by an edge. Moreover, we describe how to calculate these graphs in the case of the projective line $X = \mathbb{P}^1(\mathbb{F}_q).$