On The Cohomology of $\mathrm{SL}_2$ with Coefficients in a Simple Module

TL;DR

This paper develops a method to explicitly compute the cohomology of the group SL_2 with coefficients in simple modules, providing new formulas and confirming known results for small degrees.

Contribution

It introduces a closed form description for simple modules with non-zero cohomology and establishes a bound on the dimension of cohomology groups for SL_2.

Findings

Explicit descriptions for modules with non-zero cohomology

Confirmed cohomology results for degrees up to 3

Proposed a conjecture for general semisimple groups

Abstract

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form description of all simple modules $M$ such that $\mathrm{H}^q(G,M) \neq 0$, together with the associated dimensions $\mathrm{dim}\mathrm{H}^q(G,M)$. We apply this method for arbitrary primes $p$ and for $q \leq 3$, confirming results of Cline and Stewart along the way. Furthermore, we show that under the hypothesis $p > q$, the dimension of the cohomology $\mathrm{H}^q(G,M)$ is at most 1, for any simple module $M$. Based on this evidence we discuss a conjecture for general semisimple algebraic groups.