On normalizers of maximal tori in classical Lie groups

Abstract

In general, the normalizer N G ( H G ) N_G(H_G) of a maximal torus H G H_G in a semisimple complex Lie group G G does not admit a presentation as a semidirect product of H G H_G and the corresponding Weyl group W G W_G . Meanwhile, such splitting exists for classical groups corresponding to the root systems A ℓ A_\ell , B ℓ B_\ell , D ℓ D_\ell . For the remaining classical groups corresponding to the root systems C ℓ C_\ell , there still exists an embedding of the Tits extension W G T W^T_G of W G W_G into the normalizer N G ( H G ) N_G(H_G) . Here an explicit unified construction is proposed for the lifts of the Weyl groups into normalizers of maximal tori for general linear and orthogonal Lie groups via embedding into the general linear groups. For the symplectic group S p 2 ℓ \mathrm {Sp}_{2\ell } , an explanation of the impossibility of embedding W S p 2 ℓ W_{\mathrm {Sp}_{2\ell }} into the normalizer N S p 2 ℓ ( H S p 2 ℓ ) N_{\mathrm {Sp}_{2\ell }}(H_{\mathrm {Sp}_{2\ell }}) is suggested. Also, explicit formulas are computed for the adjoint action of the lifts of the Weyl groups on g = L i e ( G ) \mathfrak {g}=\mathrm {Lie}(G) . Finally, examples of Weyl groups lifts are given for the groups closely associated with classical Lie groups.