On splitting of the normalizer of a maximal torus in symplectic groups

A. Galt

Let \(\overline{G}\) be a simple connected linear algebraic group over an algebraically closed field \(\overline{\hspace{.3pt}\mathbb{F}}_p\) of positive characteristic \(p\) and let \(\sigma\) be a Steinberg endomorphism. There is a \(\sigma\)-invariant maximal torus \(\overline{T}\) of \(\overline{G}\), and all maximal tori are conjugate to \(\overline{T}\) in \(\overline{G}\). Moreover, the quotient \(N_{\overline{G}}(\overline{T})/\overline{T}\) is isomorphic to the Weyl group of \(\overline{G}\).

Problem 1. Describe the groups \(\overline{G}\) such that \(N_{\overline{G}}(\overline{T})\) splits over \(\overline{T}\).

A similar question arises in finite groups of Lie type. Namely, let \(T=\overline{T}\cap G\) be a maximal torus of \(G\), and let \(N=N_{\overline{G}}(\overline{T})\cap G\) be the algebraic normalizer. Notice that \(N\leqslant N_G(T)\), but the equality does not hold, in general.

Problem 2. Describe the groups \(G\) and their tori \(T\) such that \(N\) splits over \(T\).

We obtain answers to these problems for symplectic groups.

See also the author's pdf version: