On the smoothness of normalisers and the subalgebra structure of modular Lie algebras

Abstract

We provide results on the smoothness of normalisers in connected reductive algebraic groups G over fields k of positive characteristic p. Specifically we we give bounds on p which guarantee that normalisers of subalgebras of g in G are smooth, i.e. so that the Lie algebras of these normalisers coincide with the infinitesimal normalisers.One of our main tools is to exploit cohomology vanishing of small dimensional modules. Along the way, we obtain complete reducibility results for small dimensional modules in the spirit of similar results due to Jantzen, Guralnick, Serre and Bendel-Nakano-Pillen.1 We call a smooth algebraic group G reductive provided that Ru(G 0 ) = 1. 2 Recall that p is good if the following holds: p is not 2 if G contains a factor not of type A, p is not 3 if G contains an exceptional factor and p is not 5 if G contains a factor of type E8. The prime p is very good if it is good and it does not divide n + 1 for any factor of G of type An.