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Journal of Lie Theory 14 (2004), No. 2, 583--617
Copyright Heldermann Verlag 2004

Automorphisms of Normalizers of Maximal Tori and First Cohomology of Weyl Groups

Jean-Francois Hämmerli,
University of Lausanne, Institute for Geometry, Algebra and Topology, BCH / EPFL, 1015 Lausanne, Switzerland, jean-francois.haemmerli@ima.unil.ch

Michel Matthey
University of Lausanne, Institute for Geometry, Algebra and Topology, BCH / EPFL, 1015 Lausanne, Switzerland, michel.matthey@ima.unil.ch

Ulrich Suter
Institute for Mathematics, University of Neuchâtel, Rue Emile-Argand 11, 2007 Neuchâtel, Switzerland, ulrich.suter@unine.ch

Let T be a maximal torus in a connected compact Lie group G, and let W be the corresponding Weyl group with its natural action on T as a reflection group. The cohomology group H1(W; T) is computed for all simple Lie groups, and the general case is studied. The method is based on a suitable interpretation of H1(W; T) as a group of (outer) automorphisms of the normalizer of T.

Keywords: Cohomology of Weyl groups, compact Lie groups, normalizers of maximal tori, outer automorphisms.

MSC: 20J06, 22E15, 22D45.

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