Lifting automorphisms of quotients of adjoint representations

Let $\mathfrak g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, and let $G=G_1\times...\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i\mathfrak g_i$ where $r_i\geq 1$ for all $i$. We say that $V$ is \emph{large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank 1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $ψ$ of the quotient $Z:= V//G$ lifts to an algebraic mapping $Ψ\colon V\to V$ which sends the fiber over $z$ to the fiber over $ψ(z)$, $z\in Z$. (Most cases were already handled in [Kuttler11]). We also showed that one can choose a biholomorphic lift $Ψ$ such that $Ψ(gv)=σ(g)Ψ(v)$, $g\in G$, $v\in V$, where $σ$ is an automorphism of $G$. This leaves open the following questions: Can one lift holomorphic automorphisms of $Z$? Which automorphisms lift if $V$ is not large? We answer the first question in the affirmative and also answer the second question. Part of the proof involves establishing the following result for $V$ large. Any algebraic differential operator of order $k$ on $Z$ lifts to a $G$-invariant algebraic differential operator of order $k$ on $V$. We also consider the analogues of the questions above for actions of compact Lie groups.

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