Cohomology of modules for algebraic group sand exceptional isogenies

Abstract

For a simple, simply connected algebraic group G over an algebraically closed field k of characteristic p > 0, consider a a surjective endomorphism σ : G → G such that the fixed-point set G(σ) is a Suzuki or Ree group. Further, write Gσ to denote the schemetheoretic kernel of σ. Then, by utitlizing results of Jantzen and Bendel–Nakano–Pillen, we are able to compute the first cohomology for the Frobenius kernels with coefficients in the induced modules, H1 (Gσ, H 0 (λ)), and extensions Ext1 Gσ (L(λ), L(µ)) between the simple modules. When G(σ) is a Ree group of type F4, these results can be used to improve the known bounds for identifying extensions of simple modules in defining characteristic Ext1 G(σ) (L(λ), L(µ)) with those for the algebraic group.