Lifting automorphisms of quotients of adjoint representations

Let ${\mathfrak{g}}_i$ be a simple complex Lie algebra, $1\le i\le d$, and let $G=G_1\times \mathrm{.}\mathrm{.}\mathrm{.}\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i{\mathfrak{g}}_i$ where $r_i\ge 1$ for all $i$. We say that $V$ is \emph{large} if all $r_i\ge 2$ and $r_i\ge 3$ if $G_i$ has rank

1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $\psi$ of the quotient $Z:=V\mathrm{/}\mathrm{/}G$ lifts to an algebraic mapping $\Psi\colon V\to V$whichsendsthefiberover$z$tothefiberover$\psi(z)$,$z\in Z$. (Most cases were already handled in [Kuttler11]). We also showed that one can choose a biholomorphic lift $\mathrm{\Psi }$ such that $\mathrm{\Psi }(gv)=\sigma (g)\mathrm{\Psi }(v)$, $g\in G$,$v\in V$,where$\sigma$isanautomorphismof$G$. 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.