A note on the modality of parabolic subgroups

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 on which a family of R-orbits on X depends upon. Let G be a simple algebraic group defined over an algebraically closed field K of characteristic 0. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical P u via conjugation. The modality of P is defined as mod P ≔ mod( P : P u ). Let r and s be the semisimple rank of G and P respectively. We show that there is a quadratic polynomial ƒ with rational coefficients such that the modality of P is at least ƒ(r − s). In particular, the modality of a Borel subgroup B of G grows at least quadratically with r. As a consequence, we obtain a finiteness result for algebraic groups from [8]: there is only a finite number of simple algebraic groups admitting parabolic subgroups of prescribed semisimple rank and prescribed modality. Combining our lower bounds with upper bounds from [6], we can compute the modality of Borel subgroups in some small rank cases.