The cohomology in degree \(1\) of the group \(F_ 4\) in characteristic \(2\) with coefficients in a simple module (Q1325068)

Let \(G\) denote an algebraic group of type \(F_ 4\). In the case where \(G\) is defined over the algebraic closure of the field with two elements the author computes here the first Hochschild cohomology groups of \(G\) with coefficients in simple modules. For this he uses Jantzen's sum formula for Weyl modules (it is of course no small satisfaction to point out that the form he needs of this formula is the strengthened version due to the reviewer), a detailed analysis of tensor products of simple modules and results of \textit{E. Cline, B. Parshall, L. Scott} and \textit{W. van der Kallen} [Invent. Math. 39, 143-163 (1977; Zbl 0336.20036)] on the restrictions from \(G\)-cohomology to \(G(n)\)-cohomology (here \(G(n)\) denotes the finite group consisting of the \({\mathbf F}_{2^ n}\)-points of \(G\)). Recently the author has generalized his results by determining all extensions between simple modules for \(G\) (and a few other groups) [see J. Algebra 170, 1011-1034 (1994)].