Volume 12, issue 3 (2008)

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

Kasper K S Andersen and Jesper Grodal

Geometry & Topology 12 (2008) 1427–1460
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.

Keywords
$p$-compact group, root datum, maximal torus normalizer
Mathematical Subject Classification 2000
Primary: 55R35
Secondary: 20G99, 22E15, 55P35
Publication
Received: 11 January 2007
Revised: 1 April 2008
Accepted: 30 November 2007
Published: 10 June 2008
Proposed: Haynes Miller
Seconded: Paul Goerss, Ralph Cohen
Authors
 Kasper K S Andersen Department of Mathematical Sciences University of Aarhus DK-8000 Aarhus C Denmark http://person.au.dk/en/kksa@imf.au.dk Jesper Grodal Department of Mathematical Sciences University of Copenhagen DK-2100 Copenhagen Denmark http://www.math.ku.dk/~jg