On normalizers of maximal tori in classical Lie groups

TL;DR

This paper investigates the structure of normalizers of maximal tori in classical Lie groups, providing explicit constructions for certain types and explaining embedding limitations in symplectic groups.

Contribution

It offers a unified explicit construction of Weyl group lifts into normalizers for classical Lie groups of types A, B, D, and explains embedding obstructions for type C.

Findings

Explicit lifts of Weyl groups constructed for types A, B, D.

Embedding of Weyl groups into symplectic groups shown to be impossible.

Formulas for the adjoint action of these lifts are provided.

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.