A remark on decomposing the canonical representation of the Drinfeld curve

Recently, by studying an explicit basis, Köck and Laurent give the decomposition of the ${\bar{\mathbb{F}}}_q[{\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q)]$-module of holomorphic forms on the Drinfeld curve. We present a crystalline cohomological proof of a weaker version of this result, without specifying a basis. As a by-product we observe a similar decomposition for the Gelfand–Graev representations.