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

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 Pu and via the adjoint action on u, the Lie algebra of Pu. More generally, we consider the action of P on the lth member of the descending central series of u, denoted by (l)u. Let ℓ( u) denote the nilpotency class of Pu. In our main result we show that P acts on (l)u with a finite number of orbits precisely when ℓ( u) ≤ 4 for l = 0, or ℓ( u) ≤ 5 + 2l for l ≥ 1. Moreover, in case the field is algebraically closed, we consider the modality mod(P : (l)u) of the action of P on (l)u. We show that mod(P : (l)u grows linearly in the minimal cases which admit infinitely many orbits (i.e., ℓ( u) = 5 for l = 0, or ℓ( u) = 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 (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.