Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$.
Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus $g$ over $\overline{\mathbb F}_p$ with Newton polygon $\Delta$. You can add some level structure to make it a variety if you'd like.
Because the Newton polygon is invariant under isogeny, the Hecke correspondences preserve $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ inside $\mathcal A_{g, \overline{\mathbb F}_p}$. So Hecke operators act on the cohomology $H^* ( \mathcal A_{g, \overline{\mathbb F}_p, \Delta} )$. They commute, so the cohomology is an extension of $1$-dimensional characters of the Hecke algebra. Do these characters arise from automorphic forms on some group?
I think we know this is true for the cohomology of the full moduli space $\mathcal A_g$ because we can lift to characteristic $0$ and write the Hecke eigenclasses as eigenforms using de Rham cohomology and construct automorphic representations from those.
I think in the $g=1$ case this is true. It's sufficient to do this for the supersingular locus, but the supersingular locus is a finite set of points in bijection with the class group of a quaternion algebra ramified at $p$ and $\infty$, which you can write as a double coset space. Cohomology is just functions on it which are automorphic forms on that quaternion algebra.
