Outer automorphisms of adjoint groups of type D and non-rational adjoint groups of outer type A

Abstract

For a classical group $G$ of type ${\mathsf{D}}_n$ over a field $k$ of characteristic different from $2$, we show the existence of a finitely generated regular extension $R$ of $k$ such that $G$ admits outer automorphisms over $R$. Using this result and a construction of groups of type $\mathsf{A}$ from groups of type $\mathsf{D}$, we construct new examples of groups of type ${}^2{\mathsf{A}}_n$ with $n\equiv 3\mathrm{mod}4$ and the first examples of type ${}^2{\mathsf{A}}_n$ with $n\equiv 1\mathrm{mod}4$ $(n\ge 5)$ that are not $R$-trivial, hence not rational (nor stably rational).