A greedy open-orbit criterion for solvable algebraic group actions, with applications to Lusztig's nilpotent varieties

Abstract

Let $G$ be a connected solvable algebraic group acting rationally on a finite-dimensional vector space $U$. Using a $G$-stable complete flag, we formulate a successive-quotient procedure that decides whether $U$ contains an open $G$-orbit. When the procedure succeeds, it constructs an open-orbit vector of minimum support cardinality and determines the image of a generic stabilizer in the maximal torus quotient. We also give an infinitesimal version detecting open separable orbits. We apply the criterion to the action of $Au{t}_Q(M)$ on $Ex{t}_Q^1(M,M{)}^{\ast }$ where $M$ is a multiplicity-free representation of a Dynkin quiver. Rigidity of the corresponding component of Lusztig's nilpotent variety is thereby reduced to a rank test together with an acyclicity condition on a graph of active extension coordinates; the connected components of the resulting forest determine the generic indecomposable decomposition. For equioriented type $A$ this yields an explicit algorithm for a family of multisegments encoded by incidence matrices, including nonregular examples with repeated beginnings or ends.