Orbit closures and invariants (Q2332869)

Let \(k\) be an algebraically closed field, and let \(G\) be a reductive group over \(k\). The main result (Theorem 1.1) of the article affirms that for a reductive subgroup \(H\subset G\), the following are equivalent: (i) \(H\) is \(G\)-completely reducible; (ii) the normalizer \(N_G(H)\) is reductive and for every affine \(k\)-variety \(X\) the natural morphism of geometric quotients \(\psi_{X, H}: X^H / N_G(H) \to X/G\) induced by the inclusion \(X^H \hookrightarrow X\) of \(H\)-fixed locus is finite. As a reminder, a subgroup \(H\) of \(G\) is said to be \(G\)-completely reducible if for every parabolic subgroup \(P \subset G\) such that \(H \subset P\), there is a Levi subgroup \(L \subset P\) such that \(H \subset L\). This important notion is due to J.-P. Serre, and can be generalized to non-connected reductive groups. The authors also give a criterion (Theorem 7.2) for \(\psi_{X, H}\) to be finite and bijective. When \(k\) is of characteristic zero, the main theorem has been proven by D. Luna using the powerful machninery of étale slices, in which case the complete reducibility in (i) is equivalent to \(H\) being reductive. The method here is uniform for all \(k\), and it shows that Serre's notion of \(G\)-complete reducibility is indeed a good substitute of linear reducibility when one wants to generalize classical results in geometric invariant theory to arbitrary characteristics.