\(G\)-endomorphisms of affine \(G\)-varieties which induce automorphisms of the invariant subrings of the coordinate rings
Let \(G\) be an algebraic group defined over \(\mathbb{C},\) and let \(X\) be an affine \(G\)-variety, \(X=\operatorname{Spec}R.\) A \(G\)-equivariant endomorphism, or \(G\)-endomorphism, \(\varphi\) of \(X\) induces a \(G\)-endomorphism \(f\) of \(R\) as well as an endomorphism \(\bar{f}\) of the invariant subalgebra \(R^{G}.\) It is known that if \(\varphi\) is an automorphism then so is \(\bar{f}.\) This paper is concerned with the converse. Suppose \(\bar{f}\) is an endomorphism of \(R^{G}\) arising from a \(G\)-endomorphism \(\varphi\) on \(X\). If \(\bar{f}\) is an automorphism, under what conditions is \(\varphi\) an automorphism? Suppose that \(G\) is unipotent and \(X\) is factorial. Then it is shown that if \(\bar{f}\) is an automorphism, then so it \(\varphi.\) This is accomplished by proving it in the case where \(G=G_{a},\) the additive group, where the \(G\)-action can be described in terms of a derivation on \(R\). The result for unipotent \(G\) follows by considering a chain of closed normal subgroups of \(G\) whose quotients are all isomorphic to \(G_{a}\). Now suppose that \(G\) is reductive. Then \(R^{G}\) is finitely generated and \(X//G=\operatorname{Spec}R^{G}.\) We will denote the induced endomorphism on \(X//G\) by \(\overline{\varphi}.\) Let \(\pi\) be the quotient map \(X\rightarrow X//G.\) Suppose \(\overline{\varphi}\) is an automorphism and there exists a closed set \(V\subset X//G\) with codim\(_{X}\pi^{-1}\left( V\right) \geq2.\) If each fiber of \(\pi\) consists of a single closed orbit, then \(\varphi\) is a \(G\) -automorphism. Essential to this result is a theorem by \textit{S. Kaliman} [Proc. Am. Math. Soc. 133, No. 4, 975--977 (2005; Zbl 1060.14002)] which states that if the restriction \(\overline{\varphi}\) of an endomorphism \(\varphi\) of an affine algebraic variety to an open set whose complement has codimension at least 2 is injective, then \(\phi\) is an automorphism. Furthermore, suppose \(X\) is normal. Then, using the notation above, if \(\pi\) contains a dense orbit and codim\(_{X}(\overline {X-\varphi(X)}) \geq2\) then \(\varphi\) is an automorphism. Examples are given in the cases where \(G\) is a semisimple algebraic group and \(X=\mathfrak{g},\) its Lie algebra with the adjoint \(G\)-action, ad well as the case where \(X\) is an irreducible affine \(G\)-cone with \(\dim X//G=1.\)
scientific article; zbMATH DE number 5118806
scientific article