The Bruhat-Chevalley order of parabolic group actions in general linear groups and degeneration for \(\Delta\)-filtered modules.

*(English)*Zbl 0953.20037Let \(V\) be a finite dimensional vector space over an algebraically closed field \(k\). Let \(P\) be a parabolic subgroup of \(\text{GL}(V)\). The authors consider the adjoint action of \(P\) on the Lie algebra \(\text{Lie}(P_u)\) of its unipotent radical \(P_u\), and, more generally, on the \(l\)th member \(\text{Lie}(P_u)^{(l)}\) of the descending central series of \(\text{Lie}(P_u)\).

All instances when \(P\) has only finitely many orbits on \(\text{Lie}(P_u)^{(l)}\) for \(l\geq 0\) are known, and in this paper, the authors give in these cases a complete combinatorial description of the closure relation, i.e., the Bruhat-Chevalley order, on the set of \(P\)-orbits on \(\text{Lie}(P_u)^{(l)}\). If \(P\) is the Borel subgroup of \(\text{GL}(V)\), this Bruhat-Chevalley poset was first determined by V. V. Kashin [in his paper Orbits of adjoint and coadjoint actions of Borel subgroups of semisimple algebraic groups, in: Problems in Group Theory and Homological Algebra, Yaroslavl’, 141-159 (1997)] (where this problem has been solved for the adjoint and coadjoint actions of Borel subgroups of all simple algebraic groups).

The authors’ approach is based on the fact that there is a canonical bijection between the set of \(P\)-orbits on \(\text{Lie}(P_u)^{(l)}\) and the set of isomorphism classes of \(\Delta\)-filtered modules of a particular dimension \(\mathbf e\) of a certain quasi-hereditary algebra \(\mathcal A(t,l)\). These isomorphism classes in turn are given by the orbits of a reductive group \(G(\mathbf e)\) on the variety \(\mathcal R(\Delta)(\mathbf e)\) of all \(\mathcal A(t,l)\)-modules with \(\Delta\)-filtration and dimension vector \(\mathbf e\). In their main result, Theorem 1.1, the authors show that provided there are only finitely many isomorphism classes of indecomposable modules in the subcategory \(\mathcal F(\Delta)(\mathbf e)\) of all \(\Delta\)-filtered \(\mathcal A(t,l)\)-modules of dimension \(\mathbf e\), the following three posets coincide: (1) the Bruhat-Chevalley order on the set of \(P\)-orbits on \(\text{Lie}(P_u)^{(l)}\); (2) the Bruhat-Chevalley order on the set of \(G(\mathbf e)\)-orbits on \(\mathcal R(\Delta)(\mathbf e)\); (3) the poset opposite to the so-called hom-order on the set of isomorphism classes of \(\mathcal F(\Delta)(\mathbf e)\). The advantage of the latter order is that it can be computed explicitly for any given finite case.

All instances when \(P\) has only finitely many orbits on \(\text{Lie}(P_u)^{(l)}\) for \(l\geq 0\) are known, and in this paper, the authors give in these cases a complete combinatorial description of the closure relation, i.e., the Bruhat-Chevalley order, on the set of \(P\)-orbits on \(\text{Lie}(P_u)^{(l)}\). If \(P\) is the Borel subgroup of \(\text{GL}(V)\), this Bruhat-Chevalley poset was first determined by V. V. Kashin [in his paper Orbits of adjoint and coadjoint actions of Borel subgroups of semisimple algebraic groups, in: Problems in Group Theory and Homological Algebra, Yaroslavl’, 141-159 (1997)] (where this problem has been solved for the adjoint and coadjoint actions of Borel subgroups of all simple algebraic groups).

The authors’ approach is based on the fact that there is a canonical bijection between the set of \(P\)-orbits on \(\text{Lie}(P_u)^{(l)}\) and the set of isomorphism classes of \(\Delta\)-filtered modules of a particular dimension \(\mathbf e\) of a certain quasi-hereditary algebra \(\mathcal A(t,l)\). These isomorphism classes in turn are given by the orbits of a reductive group \(G(\mathbf e)\) on the variety \(\mathcal R(\Delta)(\mathbf e)\) of all \(\mathcal A(t,l)\)-modules with \(\Delta\)-filtration and dimension vector \(\mathbf e\). In their main result, Theorem 1.1, the authors show that provided there are only finitely many isomorphism classes of indecomposable modules in the subcategory \(\mathcal F(\Delta)(\mathbf e)\) of all \(\Delta\)-filtered \(\mathcal A(t,l)\)-modules of dimension \(\mathbf e\), the following three posets coincide: (1) the Bruhat-Chevalley order on the set of \(P\)-orbits on \(\text{Lie}(P_u)^{(l)}\); (2) the Bruhat-Chevalley order on the set of \(G(\mathbf e)\)-orbits on \(\mathcal R(\Delta)(\mathbf e)\); (3) the poset opposite to the so-called hom-order on the set of isomorphism classes of \(\mathcal F(\Delta)(\mathbf e)\). The advantage of the latter order is that it can be computed explicitly for any given finite case.

Reviewer: Vladimir L.Popov (Wien)

##### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

20G05 | Representation theory for linear algebraic groups |

17B45 | Lie algebras of linear algebraic groups |

16W70 | Filtered associative rings; filtrational and graded techniques |

##### Keywords:

general linear groups; parabolic subgroups; Bruhat-Chevalley posets; filtered modules; quasi-hereditary algebras; adjoint actions; Lie algebras; orbits
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\textit{T. Brüstle} et al., Adv. Math. 148, No. 2, 203--242 (1999; Zbl 0953.20037)

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