A Theorem of Tits, Normalizers of Maximal Tori and Fibrewise Bousfield–Kan Completions

Abstract

We use a theorem of Tits on the presentation of the normalizer of a maximal torus of a connected compact semisimple Lie group in terms of generators and relations to give several equivalent conditions for the splitting of the associated normalizer group extension and interprete them in terms of p-adic fibrewise homotopy theory. §0. Introduction The isomorphism type of a compact connected semisimple Lie group G is completely determined by the isomorphism type of the normalizer of the maximal torus J\G) as it was shown by Curtis, Wiederhold and Williams [C-W-W]. This was generalized much later by Notbohm for any compact connected Lie group [N]. In their classical paper Curtis, Wiederhold and Williams also studied the related question when the group extension 0 -» T(G) -»N(G) ^ W(G) -» I is a split extension and the normalizer N(G) is completely determined by the action of the Weyl group W(G) on the maximal torus T\G Using a theorem of Tits [T2] giving an explicit description of the normalizer in terms of generators and relations, they could decide case-by-case for which simple Lie groups the above normalizer sequence is split exact. In this note we use the theorem of Tits to interprete the splitting of the above normalizer group extension in terms of fibrewise p-adic Bousfield-Kan completion [B-K] of the associated fibration of classifying spaces. It turns Communicated by K. Saito, November 9, 1999. 1991 Mathematics Subject Classification. 20 F 55, 20 G 20, 20 J 06, 22 E 15, 57 T 10 Mathematisches Institut, University of Gottingen, Bunsenstr. 3-5, D-37073 Gottingen, Germany e-mail: neumann® cfgauss.uni-math.gwdg.de