$U$-Operators Acting on Harmonic Cocycles for $\mathrm{GL}_3$ and Their Slopes

In this article, we describe a computational study of the action of the two natural $U$-operators acting on $\Gamma$-invariant spaces of harmonic cocycles for $\mathrm{GL}_3$ for certain congruence subgroups $\Gamma$, in a positive characteristic setting. The cocycle spaces we consider are conjecturally isomorphic to spaces of Drinfeld cusp forms of rank $3$ and level $\Gamma$ via an analogue of Teitelbaum's residue map. We give explicit descriptions of the spaces of harmonic cocycles as subspaces of the vector space of coefficients, and of the resulting $U$- and Hecke operators acting on these. We then implement these formulas in a computer algebra system. Using the resulting data of slopes (and characteristic polynomials) for the Hecke actions, we observe several patterns and interesting phenomena present in our slope tables. This appears to be the first such study in a $\mathrm{GL}_3$ setting.