Outer automorphisms of adjoint groups of type 𝖣 and nonrational adjoint groups of outer type 𝖠

Abstract

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