Galois cohomology of real semisimple groups

TL;DR

This paper provides a combinatorial description of the first Galois cohomology sets for all inner forms of real semisimple groups using Kac diagrams, including explicit computations for certain complex types.

Contribution

It introduces a new combinatorial approach using Kac diagrams to compute Galois cohomology for real semisimple groups, extending previous known cases.

Findings

Explicit computation of H^1 for all real forms of type E_7

Explicit computation of H^1 for half-spin groups of type D_{2k}

Development of a combinatorial method for Galois cohomology calculations

Abstract

Let G be a connected, compact, semisimple algebraic group over the field of real numbers R. Using Kac diagrams, we describe combinatorially the first Galois cohomology sets H^1(R,H) for all inner forms H of G. As examples, we compute explicitly H^1 for all real forms of the simply connected group of type E_7 (which has been known since 2013) and for all real forms of half-spin groups of type D_{2k} (which seems to be new).