A short proof of Timashev's theorem on the real component group of a real reductive group

TL;DR

This paper provides a concise proof of Timashev's theorem, which determines the structure of the real component group of a connected reductive real algebraic group using classical results from Cartan, Matsumoto, and Casselman.

Contribution

It offers a simplified proof of Timashev's theorem by leveraging existing foundational results, clarifying the computation of the real component group.

Findings

The real component group a0_0 G(R) is explicitly characterized.

The proof connects classical results to modern understanding of real reductive groups.

Simplifies the understanding of the structure of real algebraic groups.

Abstract

Using results of Cartan, Matsumoto, and Casselman, we give a short proof of Timashev's theorem computing the real component group \pi_0 G(R) of a connected reductive real algebraic group G in terms of a maximal torus of G containing a maximal split torus.