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Let $\mathfrak g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, andlet $G=G_1\times...\times G_d$ 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 saythat $V$ is \emph{large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism$\psi$ of the quotient $Z:= V//G$ lifts to an algebraic mapping $\Psi\colonV\to V$ which sends the fiber over $z$ to the fiber over $\psi(z)$, $z\in Z$.(Most cases were already handled in [Kuttler11]). We also showed that one canchoose a biholomorphic lift $\Psi$ such that $\Psi(gv)=\sigma(g)\Psi(v)$, $g\inG$, $v\in V$, where $\sigma$ is an automorphism of $G$. This leaves open thefollowing questions: Can one lift holomorphic automorphisms of $Z$? Whichautomorphisms lift if $V$ is not large? We answer the first question in theaffirmative and also answer the second question. Part of the proof involvesestablishing the following result for $V$ large. Any algebraic differentialoperator of order $k$ on $Z$ lifts to a $G$-invariant algebraic differentialoperator of order $k$ on $V$. We also consider the analogues of the questionsabove for actions of compact Lie groups.

Lifting automorphisms of quotients of adjoint representations