Quotients, automorphisms and differential operators

Let $V$ be a $G$-module where $G$ is a complex reductive group. Let $Z:=\quot VG$denotethecategoricalquotientandlet$\pi\colon V\to Z$ be the morphism dual to the inclusion $\text{O}(V{)}^G\subset \text{O}(V)$. Let $\varphi \text{\hspace{0.17em}⁣}:Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:V\to V$ which lifts $\varphi$, i.e., $\pi (\mathrm{\Phi }(v))=\varphi (\pi (v))$ for all $v\in V$. In \cite{Kuttler} the case is treated where $V=r\text{lieg}$ is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\ge 2$ will do), any $\varphi$ has a lift. We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of $Z$, and we show that if a holomorphic $\varphi$ and its inverse lift holomorphically, then $\varphi$ has a lift $\mathrm{\Phi }$ which is an automorphism such that $\mathrm{\Phi }(gv)=\sigma (g)\mathrm{\Phi }(v)$, $v\in V$, $g\in G$ where $\sigma$ is an automorphism of $G$. We reduce the lifting problem to the group of automorphisms of $Z$ which preserve the natural grading of $\text{O}(Z)\simeq \text{O}(V{)}^G$. Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler’s methods to show lifting in case $V$ contains a copy of $\text{lieg}$.