Normalizers of maximal tori and real forms of Lie groups

Abstract

For a complex reductive Lie group $G$ 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 $N_G(H)$ of the maximal torus $H\subset G$ such that $W_G^T$ together with $H$ generate $N_G(H)$. We give an interpretation of the Tits classical construction in terms of the maximal split real form $G(\mathbb{R})\subset G(\mathbb{C})$, leading to a simple topological description of $W^T_G$. We also propose a different extension $W_G^U$ of the Weyl group $W_G$ associated with the compact real form $U\subset G(\mathbb{C})$. This results into a presentation of the normalizer of maximal torus of the group extension $U\ltimes {\rm Gal}(\mathbb{C}/\mathbb{R})$ by the Galois group ${\rm Gal}(\mathbb{C}/\mathbb{R})$. We also describe explicitly the adjoint action of $W_G^T$ and $W^U_G$ on the Lie algebra of $G$.Comment: 17 page