MOP—Algorithmic Modality Analysis for Parabolic Group Actions

Abstract

Let G be a simple algebraic group and P a parabolic subgroup of G. The group P acts on the Lie algebra P u of its unipotent radical P u via the adjoint action. The modality of this action, mod (P : P u ), is the maximal number of parameters upon which a family of P-orbits on Pu depends. More generally, we also consider the modality of the action of P on an invariant subspace n of P u , that is mod (P : n). In this note we describe an algorithmic procedure, called MOP, which allows one to determine upper bounds for mod (P : n). The classification of the parabolic subgroups P of exceptional groups with a finite number of orbits on p u was achieved with the aid of MOP. We describe the results of this classification in detail in this paper. In view of the results from [Hille and Rohrle 99], this completes the classification of parabolic subgroups of all reductive algebraic groups with this finiteness property. Besides this result we present other applications of MOP, and illu strate an example.