Lifting automorphisms of quotients of adjoint representations. (Q2927929)

Let \(\mathfrak g_i\) be simple complex Lie algebras, \(1\leq i\leq d\), and let \(G=G_1\times\cdots\times G_d\) be the corresponding adjoint group. Consider the \(G\)-module \(V=\bigoplus r_ i\mathfrak g_ i\) where \(r_i\in\mathbb N\) for all \(i\). The author says that \(V\) is \(large\) if \(r_i\geq 2\) for all \(i\) and \(r_i\geq 3\) when \(G_ i\) has rank 1. Earlier [J. Lond. Math. Soc., II. Ser. 89, No. 1, 169-193 (2014; Zbl 1339.20038)], the author showed that when \(V\) is large any algebraic automorphism \(\psi\) of the quotient \(Z:=V//G\) lifts to an algebraic mapping \(\Psi\colon V\to V\) which sends the fiber over \(z\) to the fiber over \(\psi(z)\) when \(z\in Z\). Now he obtains a similar result for holomorphic \(\psi\). He determines which \(\psi\) lift if \(V\) is not large. A similar problem is considered for actions of compact Lie groups.