Outer automorphisms of algebraic groups and determining groups by their maximal tori

TL;DR

This paper establishes a cohomological criterion for the existence of outer automorphisms in semisimple algebraic groups and applies it to specific types over global fields, advancing understanding of group automorphisms and their relation to maximal tori.

Contribution

It introduces a new cohomological criterion for outer automorphisms and applies it to type D_2n groups over global fields, completing previous results and providing new proofs.

Findings

Cohomological criterion for outer automorphisms

Application to type D_2n groups over global fields

Completion of previous results on arithmetic groups

Abstract

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.