Algorithmic Modality Analysis for Parabolic Groups

Abstract

For an algebraic group R acting morphically on an algebraic variety X the modality of the action, mod (R:X), is the maximal number of parameters upon which a family of R-orbits on X depends. Let G be a reductive algebraic group defined over an algebraically closed field K. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical Pu via conjugation and on \(\mathfrak{p}_u \), the Lie algebra of Pu, via the adjoint action. The modality of P is defined as mod P:=mod (P:\(\mathfrak{p}_u \)). In this paper we discuss an algorithm which is used to compute upper bounds for mod P along with some results obtained by this algorithm. One is a classification of parabolic subgroups P of simple algebraic groups G of semisimple rank 2 and modality 0. For parabolic subgroups of semisimple rank 3 we present some partial results. This extends the results of Kashin and Popov and Rohrle, where the cases of semisimple rank 0 and 1 are handled. For exceptional groups G we show that P ⊂ G has modality zero provided the class of nilpotency of Pu is at most two. The analogous result for classical groups is proved by Rohrle. For Borel subgroups B of simple groups we are able to determine the value for mod B in some small rank cases by combining lower bounds for mod B of Rohrle with upper bounds provided by the algorithm.