Galois cohomology of real quasi-connected reductive groups

TL;DR

This paper characterizes quasi-connected reductive groups over arbitrary fields and computes their first Galois cohomology over the real numbers, linking algebraic structure to Galois cohomology via Weyl group actions.

Contribution

It provides a characterization of quasi-connected reductive groups as normal subgroups of connected reductive groups and computes their Galois cohomology explicitly over the reals.

Findings

Characterization of quasi-connected reductive groups as normal subgroups.

Explicit computation of H^1(R,G) for these groups over the reals.

Description of Galois cohomology in terms of Weyl group actions.

Abstract

By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H^1(R,G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.