We consider the action of a parabolic subgroup of the General Linear Group on
a metabelian ideal. For those actions, we classify actions with finitely many
orbits using methods from representation theory.Comment: 10 pages, 6 eps figure

Goodwin, Simon M.

Hille, Lutz

Röhrle, Gerhard

English

arXiv

Abstract. 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 d = (d1,..., dt)  ∈ Zt≥1 with ∑t i=1 di = n, 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. 1

Orbits of parabolic subgroups on metabelian ideals

Abstract

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