On the component group of a real reductive group

Abstract

For a connected linear algebraic group $G$ defined over $\mathbb{R}$, we compute the component group ${\pi }_{0}G(\mathbb{R})$ 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.

11 pages, typos corrected, examples added, references updated

References in corpus (1)

• Galois cohomology and component group of a real reductive group

Cited by in corpus (2)

• Galois cohomology and component group of a real reductive group
• Computation of the component group of an arbitrary real algebraic group