Automorphisms of root data, maximal torus normalizers, and $p$-compact groups

Abstract. We describe the outer automorphism group of a compact connected Lie group as a certain subgroup of the outer automorphism group of its maximal torus normalizer, expressed in terms of the associated root datum. The same subgroup can be defined for connected 2-compact groups. We use this to show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to an action, analogous to a classical theorem of de Siebenthal for compact Lie groups, and we find a candidate formula for the whole space of self-homotopy equivalences of any connected 2-compact group. The results of this paper play a key role in a subsequent paper by the authors where we prove the conjectured classification of 2-compact groups and describe their automorphism spaces. 1.