Normalizers of maximal tori and real forms of Lie groups

Given a complex connected reductive Lie group G with a maximal torus H ⊂ G , Tits defined an extension W G T of the corresponding Weyl group W G . The extended group is supplied with an embedding into the normalizer N G ( H ) such that W G T together with H generate N G ( H ) . In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form G ( R ) ⊂ G , which leads to a simple topological description of W G T . We also consider a variation of the Tits construction associated with compact real form U of G . In this case we define an extension W G U of the Weyl group W G , naturally embedded into the group extension U ~ : = U ⋊ Γ of the compact real form U by the Galois group Γ = Gal ( C / R ) . Generators of W G U are squared to identity as in the Weyl group W G . However, the non-trivial action of Γ by outer automorphisms requires W G U to be a non-trivial extension of W G . This gives a specific presentation of the maximal torus normalizer of the group extension U ~ . Finally, we describe explicitly the adjoint action of W G T and W G U on the Lie algebra of G .