The canonical representation of the Drinfeld curve

If $C$ is a smooth projective curve over an algebraically closed field $\mathbb{F}$ and $G$ is a subgroup of automorphisms of $C$, then $G$ acts linearly on the $\mathbb{F}$-vector space of holomorphic differentials $H^0\big(C,\Omega_C\big)$ by pulling back differentials. In other words, $H^0\big(C,\Omega_C\big)$ is a representation of $G$ over the field $\mathbb{F}$, called $\textit{the canonical representation}$ of $C$. Computing its decomposition as a direct sum of indecomposable representations is still an open problem when the ramification of the cover of curves $C \longrightarrow C/G$ is wild. In this paper, we compute this decomposition for $C$ the Drinfeld curve ${XY^q-X^qY-Z^{q+1}=0}$, $\mathbb{F}=\bar{\mathbb{F}}_q$, and ${G=SL_2\big(\mathbb{F}_q\big)}$ where $q$ is a prime power.