Finite, tame, and wild actions of parabolic subgroups in GL(V) on certain unipotent subgroups

摘要

Let P be a parabolic subgroup of some general linear group GL(V) where v is a finite-dimensional vector space over an infinite field. The group P acts by conjugation on its unipotent radical P-u and via the adjoint action on p(u) the Lie algebra of P-u. More generally, we consider the action of P on the lth member of the descending central series of p(u), denoted by p(u)((l)). Let l(p(u)) denote the nilpotency class of P,,. In our main result we show that P acts on p(u)((l)) with a finite number of orbits precisely when l(p(u)) less than or equal to 4 for l = 0, or l(p(u)) less than or equal to 5 + 21 for l greater than or equal to 1. Moreover, in case the field is algebraically closed, we consider the modality mod(P : p(u)((l))) of the action of P on p(u)((l)). We show that mod(P : p(u)((l))) grows linearly in the minimal cases which admit infinitely many orbits (i.e., l(p(u)) = 5 for l = 0, or l(p(u)) = 6 + 2l for l greater than or equal to 1), whereas the corresponding modality grows quadratically in all other infinite cases. These results are obtained by interpreting the orbits of P on p(u)((l)) as isomorphism classes of good modules over certain quasi-hereditary algebras and by a detailed inspection of the Delta-representation types of these algebras. (C) 2000 Academic Press. [References: 18]