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 $\sigma :G\to G$ 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.

42 pages, improved bounds in Theorem 4.2.5