Abstract: The action of G = SL_2(F_q) on the Drinfeld curve was first investigated by
Drinfeld in 1974. He noted the cuspidal representations (certain characteristic 0
representations) of G can be found in the first etale cohomology of this curve. Our work
pertains to decomposing the characteristic p representations that come from this action
on spaces of globally holomorphic poly-differentials. This involves utilising tools
from both Algebraic Geometry and modular representation theory, including studying
ramification, divisors, Brauer trees, and Auslander-Reiten quivers. We first compute
the decomposition for the subgroup of upper-triangular matrices of G, and then use
the Green correspondence to lift this decomposition to the full group.