Normalizers of maximal tori and real forms of Lie groups

Abstract

Given a complex connected reductive Lie group G with a maximal torus Hsubset G, Tits defined an extension W_G^{mathrm{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^{mathrm{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(mathbb {R})subset G, which leads to a simple topological description of W^{mathrm{T}}_G. 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 widetilde{U}:=U,{rtimes }, Gamma of the compact real form U by the Galois group Gamma ={mathrm{Gal}}(mathbb {C}/mathbb {R}). Generators of W^U_G are squared to identity as in the Weyl group W_G. However, the non-trivial action of Gamma by outer automorphisms requires W^U_G to be a non-trivial extension of W_G. This gives a specific presentation of the maximal torus normalizer of the group extension {widetilde{U}}. Finally, we describe explicitly the adjoint action of W_G^{mathrm{T}} and W^U_G on the Lie algebra of G.