On normalizers of maximal tori in classical Lie groups

Abstract

The normalizer $N_G(H_G)$ of a maximal torus $H_G$ in a semisimple complex Lie group $G$ does not in general allow a presentation as a semidirect product of $H_G$ and the corresponding Weyl group $W_G$. Meanwhile, splitting holds for classical groups corresponding to the root systems $A_\ell$, $B_\ell$, $D_\ell$. For the remaining classical groups corresponding to the root systems $C_\ell$ there still exists an embedding of the Tits extension of $W_G$ into normalizer $N_G(H_G)$. We provide explicit unified construction of the lifts of the Weyl groups into normalizers of maximal tori for classical Lie groups corresponding to the root systems $A_\ell$, $B_\ell$, $D_\ell$ using embeddings into general linear Lie groups. For symplectic series of classical Lie groups we provide an explanation of impossibility of embedding of the Weyl group into the symplectic group. The explicit formula for adjoint action of the lifts of the Weyl groups on $\mathfrak{g}={\rm Lie}(G)$ are given. Finally some examples of the groups closely associated with classical Lie groups are considered.