Orbits of parabolic subgroups on metabelian ideals

Let k be an algebraically closed field, t is an element of Z(>= I), and let B be the Borel subgroup of GL(t)(k) consisting of upper-triangular matrices. Let Q be a parabolic subgroup of GL(t)(k) that contains B and such that the Lie algebra q(u) of the unipotent radical of Q is metabelian, i.e. the derived subalgebra of q(u) is abelian. For a dimension vector d = (d(1), ... , d(t)) is an element of Z(>= I)(t), with Sigma(t)(i-1) d(i) = n, we obtain a parabolic subgroup P(d) of GL(n)(k) from B by taking upper-triangular block matrices with (i, j) block of size d(i) x d(j). In a similar manner we obtain a parabolic subgroup Q(d) of GL(n)(k) from Q. We determine all instances when P(d) acts on q(n)(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. (c) 2008 Elsevier Masson SAS. All rights reserved.