Orbits of parabolic subgroups on metabelian ideals

Let k be an algebraically closed field, t∈Z⩾1, and let B be the Borel subgroup of GLt(k) consisting of upper-triangular matrices. Let Q be a parabolic subgroup of GLt(k) that contains B and such that the Lie algebra qu of the unipotent radical of Q is metabelian, i.e. the derived subalgebra of qu is abelian. For a dimension vector with , we obtain a parabolic subgroup P(d) of GLn(k) from B by taking upper-triangular block matrices with (i,j) block of size di×dj. In a similar manner we obtain a parabolic subgroup Q(d) of GLn(k) from Q. We determine all instances when P(d) acts on qu(d) with a finite number of orbits for all dimension vectors d. Our methods use a translation of the problem into the representation theory of certain quasi-hereditary algebras. In the finite cases, we use Auslander–Reiten theory to explicitly determine the P(d)-orbits; this also allows us to determine the degenerations of P(d)-orbits.