A remark on decomposing the canonical representation of the Drinfeld curve

TL;DR

This paper provides a crystalline cohomological proof of a decomposition of the module of holomorphic forms on the Drinfeld curve, extending understanding of its structure without relying on explicit bases.

Contribution

It offers a new proof technique for the decomposition of the Drinfeld curve's module, avoiding explicit basis construction and revealing related decompositions for Gelfand--Graev representations.

Findings

Crystalline cohomology offers an alternative proof of the decomposition.

A similar decomposition applies to Gelfand--Graev representations.

The approach simplifies understanding of the module's structure.

Abstract

Recently, by studying an explicit basis, K\"ock and Laurent give the decomposition of the $\overline{\mathbb{F}}_q[\mathrm{SL}_2(\mathbb{F}_q)]$-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.