Finite, Tame, and Wild Actions of Parabolic Subgroups in GL(V) on Certain Unipotent Subgroups

AbstractLet 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 Pu and via the adjoint action on pu, the Lie algebra of Pu. More generally, we consider the action of P on the lth member of the descending central series of pu, denoted by p(l)u. Let ℓ(pu) denote the nilpotency class of Pu. In our main result we show that P acts on p(l)u with a finite number of orbits precisely when ℓ(pu)≤4 for l=0, or ℓ(pu)≤5+2l for l≥1. Moreover, in case the field is algebraically closed, we consider the modality mod(P:p(l)u) of the action of P on p(l)u. We show that mod(P:p(l)u grows linearly in the minimal cases which admit infinitely many orbits (i.e., ℓ(pu)=5 for l=0, or ℓ(pu)=6+2l for l≥1), whereas the corresponding modality grows quadratically in all other infinite cases. These results are obtained by interpreting the orbits of P on p(l)u as isomorphism classes of good modules over certain quasi-hereditary algebras and by a detailed inspection of the Δ-representation types of these algebras