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 σ:G→G be a surjective endomorphism of G such that the fixed point set G(σ) is a Suzuki or Ree group. Then, let Gσ denote the scheme-theoretic kernel of σ. Using methods of Jantzen and Bendel–Nakano–Pillen, we compute the 1-cohomology for the Frobenius kernels with coefficients in the induced modules, H1(Gσ,H0(λ)), and extensions between the simple modules, ExtGσ1(L(λ),L(μ)). When G(σ) is a Ree group of type F4, these results are used to improve the known bounds for identifying extensions of simple modules in defining characteristic ExtG(σ)1(L(λ),L(μ)) with those for the algebraic group.