On the cohomology of the Ree groups and kernels of exceptional isogenies

Abstract

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$. Let $\sigma : G \rightarrow G$ be a surjective endomorphism of $G$ such that the fixed point set $G(\sigma)$ is a Suzuki or Ree group. Then, let $G_{\sigma}$ denote the scheme-theoretic kernel of $\sigma.$ Using methods of Jantzen and Bendel-Nakano-Pillen, we compute the $1$-cohomology for the Frobenius kernels with coefficients in the induced modules, $H^{1}(G_{\sigma}, H^{0}(\lambda))$, and the $1$-cohomology for the Frobenius kernels with coefficients in the simple modules, $H^{1}(G_{\sigma}, L(\lambda))$ for the Suzuki and Ree groups. Moreover, we improve the known bounds for identifying extensions for the Ree groups of type $F_4$ with the ones for the algebraic group.