Twisted conjugacy in simply connected groups.
\(G\) is a simply connected semisimple group over the algebraically closed field \(k\) and \(\sigma\) is a non-trivial diagram automorphism of \(G\). One has the twisted conjugation action of \(G\) on itself \(((g,h)\mapsto gh(\sigma g)^{-1})\). This paper collects a number of results concerning this action. Some samples: The fixed point group \(G^\sigma\) is semi-simple. The geometric quotient \(G//_\sigma G\) of \(G\) by the twisted action is isomorphic to \(\mathbb{A}^d\), where \(d=\dim T^\sigma\) for a \(\sigma\)-invariant maximal torus \(T\) in a \(\sigma\)-invariant Borel subgroup \(B\). The twisted conjugates \(g\) of elements of \(T\) form a dense subset of \(G\). They have a connected \(\sigma\)-centralizer \(Z_\sigma(g)=\{x\in G\mid xg(\sigma x)^{-1}=g\}\).
scientific article; zbMATH DE number 5118045