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

Let $G$ be the simple algebraic group ${\mathrm{S}\mathrm{L}}_2$ 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 $$\mathrm{H}$^q(G,M) \neq 0$,togetherwiththeassociateddimensions$$\mathrm{d}\mathrm{i}\mathrm{m}$$\mathrm{H}$^q(G,M)$. We apply this method for arbitrary primes $p$ and for $q\le 3$, 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.