Generic stabilizers in actions of simple algebraic groups (Q6605409)

In the paper under review, the authors treat the faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties.\N\NLet \(G\) be a simple algebraic group over an algebraically closed field \(K\) of characteristic \(p\) and \(V\) a non-trivial irreducible \(G\)-module of dimension \(d\). For \(k = 1, \ldots, d\) the Grassmannian variety \(\mathcal{G}_{k}(V)\) consists of the \(k\)-dimensional subspaces of \(V\), and has dimension \(k(d-k)\); as the action of \(G\) on \(V\) is linear, it extends naturally to \(\mathcal{G}_{k}(V)\). Since \(\mathcal{G}_{d-k}(V) \simeq \mathcal{G}_{k}(V^{\ast})\) (where \(V^{\ast}\) is the dual of \(V\)), it is sufficient to consider the cases \(1 \leq k \leq d/2\). Let \(X\) be an irreducible variety on which \(G\) acts. If \(G_{X}\) is the kernel of the action, then \(G/G_{X}\) acts faithfully on \(X\). If \(\widehat{X}\) is a non-empty open set in \(X\) and for all \(x, x'\in \widehat{X}\), the stabilizers in \(G/G_{X}\) of \(x\) and \(x'\) are isomorphic subgroups, then the action has a semi-generic stabilizer, whose isomorphism type is shared by each such subgroup \(C_{G/G_{X}}(x)\) for \(x\in \widehat{X}\). If moreover \(\widehat{X}\) has the property that for all \(x,x' \in \widehat{X}\) the stabilizers in \(G/G_{X}\) of \(x\) and \(x'\) are conjugate subgroups, we say that the action has a generic stabilizer, whose conjugacy class is that containing each such subgroup \(C_{G/G_{X}}(x)\) for \(x\in \widehat{X}\).\N\NThe most basic result is Theorem 1: Let \(G\) be a simple algebraic group over an algebraically closed field of characteristic \(p\), and \(V\) a non-trivial irreducible \(G\)-module of dimension \(d\). (i) The action of \(G\) on \(V\) has a generic stabilizer. (ii) For \(1 \leq k \leq d/2\), either the action of \(G\) on \(\mathcal{G}_{k}(V)\) has a generic stabilizer, or \(G =\mathsf{B}_{3}\) or \(\mathsf{C}_{3}\), \(p=2\), \(V\) is the spin module for \(G\) and \(k=4\), in which case the action of \(G\) on \(\mathcal{G}_{k}(V)\) has a semi-generic stabilizer but not a generic stabilizer.\N\NThe proof of Theorem 1 and the determination of the generic stabilizers occupy the entirety of this work and involve a great deal of case analysis. The authors provide tables listing generic stabilizers in the cases where they are non-trivial. In addition, they decide whether or not there is a dense orbit or a regular orbit for the action on the module.