Stability of affine G-varieties and irreducibility in reductive groups (Q2909479)

The aim of this paper is to give a new criterion for orbits of a complex reductive group \(G\) on an affine variety \(X\) to be closed (a ``polystability criterion'' in the language of the authors). Recall that given a cocharacter (one-parameter subgroup) \(\lambda\) of \(G\) and a point \(x \in X\), we say that the limit \(\lim_{t\to0}\lambda(t)\cdot x\) exists if the morphism \(t\mapsto\lambda(t)\cdot x\) from \(\mathbb{C}^* \to X\) extends to a morphism from all of \(\mathbb{C}\). The set of such cocharacters for a point \(x \in X\) is denoted \(\Lambda_x\). Recall also that, in the case where \(X=G\) and \(G\) acts by conjugation, we can recover the parabolic subgroups of \(G\) from the cocharacters: for each \(\lambda\) we have \(P(\lambda)=\{g \in G \mid \lim_{t\to0}\lambda(t)g\lambda(t)^{-1}\) exists\(\}\) is a parabolic subgroup of \(G\), and all parabolics arise in this fashion. Given \(x \in X\), let \(H_x\) denote the intersection of all \(P(\lambda)\) as \(\lambda\) runs over the set \(\Lambda_x\). The main result of this paper can be stated as follows:NEWLINENEWLINESuppose the affine \(G\)-variety \(X\) has the following property: for all cocharacters \(\lambda\) and \(x \in X\), if \(H_x \subseteq P(\lambda)\), then \(\lambda \in \Lambda_x\). Then the \(G\)-orbit of \(x\) is closed in \(X\) if and only if \(H_x\) is a reductive (not necessarily connected) subgroup of \(G\).NEWLINENEWLINEThe authors are motivated by the case that \(X=G^n\) is the direct product of \(n\) copies of \(G\) and \(G\) acts by simultaneous conjugation, in which case the hypotheses of the above result hold. This action was extensively studied by \textit{R.W. Richardson} [Duke Math. J. 57, No. 1, 1--35 (1988; Zbl 0685.20035)], and the authors reprove some of his results, which easily extend to \(G\)-stable subvarieties of \(G^n\) such as representation varieties. It is not clear that there are other interesting classes of variety satisfying the hypotheses of the main result. The authors also phrase their results in terms of the subgroup \(H_x\) being \(G\)-completely reducible in the sense of \textit{J-P. Serre} [Astérisque 299, 195--217, Exp.~No.~932 (2005; Zbl 1156.20313)], which is not particularly useful in characteristic zero because a subgroup of \(G\) is \(G\)-completely reducible if and only if it is reductive. However, this does suggest that they could have proved their results in positive characteristic using this notion. There is no mention of this possibility in the paper.