Lifting automorphisms of generalized adjoint quotients (Q413944)

Let \(G\) be a reductive algebraic group over an algebraically closed field of zero characteristic and let \(\mathfrak g\) be the Lie algebra of \(G\) endowed with the adjoint action of \(G\). Consider the diagonal action of \(G\) on the product \(\mathfrak g^r\) of \(r\) copies of \(\mathfrak g\) and let \(\pi: \mathfrak g^r\to X:=\mathfrak g^r/\!\!/G\) be the categorical quotient for this action. The author proves that if \(r\) is suitably large (in general, \(r\geqslant 5\), but in many cases \(r\geqslant 3\) or even \(r\geqslant 2\) suffices), then every automorphism \(\varphi: X\to X\) lifts to a map \(\Phi: \mathfrak g^r\to \mathfrak g^r\) such that \(\pi\circ\Phi=\varphi\circ\pi\). An application of this result to the Luna stratification is given.