Orbits of parabolic subgroups on metabelian ideals

Abstract

Let k be an algebraically closed field, t ∈ Z ⩾ 1 , 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 ) ∈ Z ⩾ 1 t with ∑ i = 1 t 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 × 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 u ( 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.