Twisted linear actions on complex Grassmannians (Q1190704)

In this paper, the author studies some twisted actions of Lie groups on complex Grassmannians. Such an action is associated to a representation of a Lie group \(\rho: G\to Gl(n,\mathbb{C})\) and a square matrix \(X\in M_ n(\mathbb{C})\), such that \(X\) has eigenvalues with positive real parts and \(\rho(g)X=X\rho(g)\) for every \(g\in G\). He defines a notion of equivalence of such actions and proves the following: 1) For \(G\) compact, every pair \((\rho,X)\) is equivalent to a pair \((\rho,I_ n)\) where \(I_ n\) is the identity matrix. 2) For \(G=Sl(n,\mathbb{C})\), the author gives examples of uncountably many actions.