On a question of Kulshammer for representations of finite groups in reductive groups

摘要

Let G be a simple algebraic group of type G (2) over an algebraically closed field of characteristic 2. We give an example of a finite group I" with Sylow 2-subgroup I"(2) and an infinite family of pairwise non-conjugate homomorphisms rho: I" -> G whose restrictions to I"(2) are all conjugate. This answers a question of Burkhard Kulshammer from 1995. We also give an action of I" on a connected unipotent group V such that the map of 1-cohomologies H (1)(I", V) -> H (1)(I" (p) , V) induced by restriction of 1-cocycles has an infinite fibre.