Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras (Q321887)

Let \(G\) be a simple algebraic group over a field \(F\). The inner automorphisms of \(G\) are the \(F\)-points of \(\Aut(G)^\circ\), the identity component of the algebraic group \(\Aut(G)\). The authors consider the question of existence of outer automorphisms for \(G\), focusing on the cases where \(G\) is of type \({}^3D_4\) or \({}^2A_n\) (\(n\) even).NEWLINENEWLINEThe question is refined as follows: Let \(\Delta\) denote the Dynkin diagram of \(G\) over \(F_{\mathrm{sep}}\). There is an exact sequence of groups NEWLINE\[NEWLINE1\to \Aut(G)^\circ(F)\to \Aut(G)(F)\rightarrow{\alpha} \Aut(\Delta)(F)\;,NEWLINE\]NEWLINE where the last term denotes the \(\mathrm{Gal}(F_{\mathrm{sep}}/F)\)-invariant automorphisms of \(\Delta\). Thus, one can ask whether an element of \(\Aut(\Delta)(F)\) comes from an automorphism of \(G\), i.e.\ whether \(\alpha\) is onto. This is false in general, and can be observed using a finer obstruction: Let \(Z\) denote the scheme-theoretic center of the simply connected cover of \(G\). Then \(G\) gives rise to an element \(t_G\in\mathrm{H}^2_{\text{fppf}}(F,Z)\) called the \textit{Tits class} of \(G\) (see \(\S 31\) in [\textit{M.-A. Knus} et al., The book of involutions. Providence, RI: American Mathematical Society (1998; Zbl 0955.16001)]). The group \(\Aut(\Delta)(F)\) acts on \(\mathrm{H}^2_{\text{fppf}}(F,Z)\) and in [\textit{S. Garibaldi}, Mich. Math. J. 61, No. 2, 227--237 (2012; Zbl 1277.20057)] it was shown that NEWLINE\[NEWLINE\mathrm{im}(\alpha)\subseteq \{\pi\in \Aut(\Delta)(F)\,:\,\pi(t_G)=t_G\}\;.NEWLINE\]NEWLINE One can therefore ask whether an equality holds. While this is still false in general, the authors conjecture that equality holds when \(G\) is absolutely simple, and simply connected or adjoint. Indeed, the conjecture is known to hold when \(G\) is of inner type or of type \({}^6D_4\) (Example 17 in [Garibaldi, loc. cit.]).NEWLINENEWLINEThe main result of the paper establishes of the conjecture for types \({}^3D_4\) and \({}^2A_n\) (\(n\) even), where almost all of the work concerns with the former case. To show it, the authors prove a Skolem-Noether theorem for Albert algebras: Let \(E\) and \(E'\) be cubic étale subalgebras of an Albert \(F\)-algebra \(J\) and let \(\varphi : E\to E'\) be an isomorphism. Then there exists \(w\in E\) of norm \(1\) such that the map \(e\mapsto \varphi(ew):E\to E'\) extends to an \textit{isotopy} of \(J\). This holds when \(F\) has arbitrary characteristic. Similar results for absolutely simple \(9\)-dimensional Jordan algebras are also obtained.NEWLINENEWLINEOne can further ask whether an element \(\pi\) in the image of \(\alpha\) can be lifted to an automorphism of the same order as \(\pi\). This is shown to hold when \(G\) has type \({}^2A_n\) (\(n\) even). However, basing on [\textit{M.-A. Knus} and \textit{J.-P. Tignol}, Doc. Math., J. DMV Extra Vol., 387--405 (2015; Zbl 1365.20048)], the authors exhibit a counterexample for type \({}^3 D_4\). This seems to be the first known example where \(\alpha:\Aut(\Delta)(F)\to \{\pi\in \Aut(\Delta)(F)\,:\,\pi(t_G)=t_G\}\) is onto but not split.