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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 $\\psi$ 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 $\\psi(z)$, $z\\in Z$. (Most cases were already handled in [Kuttler11]). 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Schwarz","submitted_at":"2013-01-26T23:53:41Z","abstract_excerpt":"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 $\\psi$ 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 $\\psi(z)$, $z\\in Z$. (Most cases were already handled in [Kuttler11]). 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