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

TL;DR

This paper computes the first cohomology groups of Frobenius kernels associated with Suzuki and Ree groups, providing new insights into their structure and extending existing bounds for Ree groups of type F4.

Contribution

It introduces explicit calculations of first cohomology groups for Frobenius kernels of Suzuki and Ree groups, enhancing understanding of their extension properties.

Findings

Computed H^1 for Frobenius kernels with induced modules

Computed H^1 for Frobenius kernels with simple modules

Improved bounds for extensions of Ree groups of type F4

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.