On the component group of a real reductive group

TL;DR

This paper computes the component group of a real reductive algebraic group using maximal split tori, providing explicit representatives for each connected component and extending classical results with new cohomological methods.

Contribution

It explicitly determines the component group of real reductive groups in terms of split tori and offers concrete representatives for all components, building on Galois cohomology techniques.

Findings

Explicit description of the component group $\

All connected components intersect the maximal split torus $\

Provides explicit elements representing each component.

Abstract

For a connected linear algebraic group $G$ defined over $\mathbb{R}$, we compute the component group $\pi_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ in terms of a maximal split torus $T_{\text{s}}\subseteq G$. In particular, we recover a theorem of Matsumoto (1964) that each connected component of $G(\mathbb{R})$ intersects $T_{\text{s}}(\mathbb{R})$. We provide explicit elements of $T_{\text{s}}(\mathbb{R})$ which represent all connected components of $G(\mathbb{R})$. The computation is based on structure results for real loci of algebraic groups and on methods of Galois cohomology.