Normalizers of maximal tori and real forms of Lie groups

Abstract

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