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

TL;DR

This paper demonstrates the existence of certain outer automorphisms for classical groups of type D over specific extensions, leading to new examples of non-rational groups of outer type A.

Contribution

It constructs new non-rational groups of type ^2A_n with specific properties, expanding understanding of automorphisms and rationality in algebraic groups.

Findings

Existence of outer automorphisms over finitely generated regular extensions.

Construction of non-rational groups of type ^2A_n for specific n.

First examples of non-rational groups of type ^2A_n with certain congruence conditions.

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 $n\equiv 3\bmod 4$ and the first examples of type $^2\mathsf A_n$ with $n\equiv 1\bmod 4$ $(n\geq 5)$ that are not $R$-trivial, hence not rational (nor stably rational).