Mathematics – Group Theory
Scientific paper
2012-01-30
Mathematics
Group Theory
17 pages, corrected several errors
Scientific paper
Let $V$ be a $G$-module where $G$ is a complex reductive group. Let $Z:=V//G$ denote the categorical quotient and let $\pi\colon V\to Z$ be the morphism dual to the inclusion $O(V)^G\subset O(V)$. Let $\phi\colon Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic automorphism $\Phi\colon V\to V$ which lifts $\phi$, i.e., $\pi(\Phi(v))=\phi(\pi(v))$ for all $v\in V$. If so, can we choose $\Phi$ to have some kind of equivariance property? In Kuttler\cite{Kuttler} the case is treated where $V=k\lieg$ is a multiple of the adjoint representation of $G$. It is shown that, for $k$ sufficiently large (often $k\geq 2$ will do), any $\phi$ has a lift. We consider the case of general representations. It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of $Z$, and we show that if a holomorphic $\phi$ lifts holomorphically, then it has a lift $\Phi$ such that $\Phi(gv)=\sigma(g)\Phi(v)$, $v\in V$, $g\in G$ where $\sigma$ is an automorphism of $G$. Lifting does not always hold, but we show that it always does for representations of tori. We extend Kuttler's methods to show lifting in case $V$ contains a copy of $\lieg$.
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