Automorphisms of $p$–compact groups and their root data

Abstract

We construct a model for the space of automorphisms of a connected $p$–compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a $p$–compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ‘The classification of 2-compact groups’ [arXiv:math.AT/0611437], where we prove the conjectured classification of $2$–compact groups and determine their automorphism spaces.