On algebraic normalisers of maximal tori in simple groups of Lie type (Q6634502)

Let \(\overline G\) be an adjoint simple (affine) algebraic group over the algebraic closure \(k={\overline F}_p\) of the field \(F_p\) of prime order \(p\). Let \(\sigma\) be an endomorphism of \(\overline G\) such that the group \({\overline G}_{\sigma}\) of \(\sigma\)-stable elements of \(\overline G\) is finite. Let \(G\) be the commutator subgroup of \(O^{p'}({\overline G}_{\sigma})\). Then \(G\) is a finite simple group of Lie type. Let \(\overline T\) be a maximal \(\sigma\)-stable (that is, \({\overline T}^{\sigma}=\overline T\)) torus of \(\overline G\) and \(\overline N=N_{\overline G}(\overline T)\). Then \(T:= \overline T\cap G\) is a maximal torus of \(G\) and \(N(G, T)= \overline N\cap G\) is the algebraic normaliser of \(T\) in \(G\).\N\NIn this article, the author explicitly identifies all the cases when \(N(G, T)\) is not equal to \(N_G(T)\). For classical groups \(G\), there are several infinite series of \({\overline G}_{\sigma}\)-classes of maximal tori such that \(N(G, T)\not= N_G(T)\). They occur if \(G\) is \(\mathrm{PSL}_n(q)\), \(\Omega^{\pm}_{2n}(2)\) or \(\mathrm{PSp}_{2n}(q)'\) with \(q\in\{2,3\}\). There are exactly \(140\) such classes for exceptional groups \(G\). Note that earlier \textit{T. A. Springer} and \textit{R. Steinberg} [Lect. Notes Math. 131, E1--E100 (1970; Zbl 0249.20024)] considered the same problem for the more general case of reductive algebraic groups.