On normalizers of maximal tori in classical Lie groups

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 $W_G$. Such 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 exists an embedding of the Tits extension of $W_G$ into the normalizer $N_G(H_G)$. We provide an explicit construction of the lifts of the Weyl groups into normalizers of maximal tori for classical Lie groups $GL_{\ell+1}$ and $O_{\ell+1}$ using embeddings into general linear Lie groups. This provides an explicit description of the normalizers $N_G(H_G)$ for the general linear and orthogonal Lie groups. For symplectic series of classical Lie groups we explain impossibility of embedding of the Weyl group into the symplectic group. We also provide explicit formulas 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.