Normalizers and self-normalizing subgroups II

Let \mathbbK\mathbb{K} be a field, G a reductive algebraic \mathbbK\mathbb{K}-group, and G ₁ ≤ G a reductive subgroup. For G ₁ ≤ G, the corresponding groups of \mathbbK\mathbb{K}-points, we study the normalizer N = N _G (G ₁). In particular, for a standard embedding of the odd orthogonal group G ₁ = SO(m, \mathbbK\mathbb{K}) in G = SL(m, \mathbbK\mathbb{K}) we have N ≅ G ₁ ⋊ μ _m ( \mathbbK\mathbb{K}), the semidirect product of G ₁ by the group of m-th roots of unity in \mathbbK\mathbb{K}. The normalizers of the even orthogonal and symplectic subgroup of SL(2n, \mathbbK\mathbb{K}) were computed in Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, \mathbbK\mathbb{K}) and G ₁ = O(m, \mathbbK\mathbb{K}) we have N ≅ G ₁ ⋊ \mathbbK\mathbb{K} ^×. In both of these cases, N is a self-normalizing subgroup of G.