Automorphic functions for nilpotent extensions of curves over finite fields

Abstract

We define and study the subspace of cuspidal functions for $G$-bundles on a class of nilpotent extensions $C$ of curves over a finite field. We show that this subspace is preserved by the action of a certain noncommutative Hecke algebra ${\mathcal{H}}_{G,C}$. In the case $G={\mathrm{GL}}_2$, we construct a commutative subalgebra in ${\mathcal{H}}_{G,C}$ of Hecke operators associated with simple divisors. In the case of length 2 extensions and of $G={\mathrm{GL}}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. In this case we also construct some Hecke eigenfunctions using the relation to Higgs bundles over the corresponding reduced curve.

62 pages