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

Abstract

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 $\Gamma$ with Sylow $2$-subgroup ${\Gamma }_2$ and an infinite family of pairwise non-conjugate homomorphisms $\rho :\Gamma \to G$ whose restrictions to ${\Gamma }_2$ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of $\Gamma$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${\mathrm{H}}^1(\Gamma ,V)\to {\mathrm{H}}^1({\Gamma }_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.

5 pages. Minor changes from previous version

References in corpus (1)

• 1998 Moursund Lectures at the University of Oregon