The Bruhat–Chevalley Order of Parabolic Group Actions in General Linear Groups and Degeneration for Δ-Filtered Modules

Abstract

dedicated to helmut lenzing on the occasion of his 60th birthday Let k be an algebraically closed field and V a finite dimensional k-space. Let GL(V ) be the general linear group of V and P a parabolic subgroup of GL(V ). Now P acts on its unipotent radical P u and on p u =Lie P u , the Lie algebra of P u , via the adjoint action. More generally, we consider the action of P on the l th member of the descending central series of p u denoted by p (l ) u . All instances when P acts on p (l ) u for l 0 with a finite number of orbits are known. In this note, we give a complete combinatorial description of the closure relation on the set of P-orbits on p (l ) u , i.e., the Bruhat Chevalley order, for every finite case. There is a canonical bijection between the set of P-orbits on p (l ) u and the set of isomorphism classes of 2-filtered modules of a particular dimension vector e of a certain quasi-hereditary algebra A(t, l ). These isomorphism classes in turn are given by the orbits of a reductive group G(e) on the variety R(2)(e) of all A(t, l )-modules with 2-filtration Article ID aima.1999.1851, available online at http:ÂÂwww.idealibrary.com on 203 and dimension vector e. The subcategory of A(t, l )-mod of all 2-filtered A(t, l )modules of dimension vector e is denoted by F(2)(e). In our chief result, Theorem 1.1, we show that provided there is only a finite number of isomorphism classes of indecomposable modules in F(2) the following three posets coincide:(1) the Bruhat Chevalley order on the set of P-orbits on p (l ) u ; (2) the Bruhat Chevalley order on the set of G(e)-orbits on R(2)(e);(3) the poset opposite to the so called hom-order on the set of isomorphism classes of F(2)(e).The advantage of this hom-order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case. Theorem 1.1 then in turn allows us to explicitly determine the closure relations for the P-orbits on p (l ) u with the aid of this hom-order. We present some examples of Hasse diagrams in an Appendix. Academic PressAlthough the motivation and the goal for this work is the combinatorial description of the Bruhat Chevalley order of the parabolic group actions described above, the representation theoretic methods developed for this purpose, especially the description of degenerations of 2-filtered modules in the 2-finite setting in terms of the so called hom-order, seems to be of interest in its own right. The development of the relevant theory in Sections 3, 4, and 5, carried out for an arbitrary 2-finite quasi-hereditary algebra, may be viewed and read independently from the other parts of this work. 204BRU STLE ET AL.

Orbits of parabolic subgroups on metabelian ideals

“…In the case G = GL n (k), there has been much success in understanding the adjoint action of a parabolic subgroup through a translation in to the representation theory of certain quasi-hereditary algebras. This translation was first observed in [8], and has subsequently been further exploited, see for example, [2] and [4]. We refer the reader to [7] and [11] for recent related developments.…”

“…Remark 1.3. Using the results from [4] and the Auslander-Reiten quivers that we have calculated, it is possible to calculate the degenerations of P (d)-orbits in q u (a, d) in the finite cases. More precisely, let O, O ′ be P -orbits and M, M ′ the corresponding A(a)-modules.…”

“…More precisely, let O, O ′ be P -orbits and M, M ′ the corresponding A(a)-modules. In [4] it is shown that M is greater than M ′ in the hom-order if and only if O is a degeneration of O ′ . The hom-order can be read off from the Auslander-Reiten quiver.…”

On the Complement of the Dense Orbit for a Quiver of Type 𝔸

“…We explain the common idea and some generalizations. Note that degenerations have been already studied in [5].…”

Semisimplification for subgroups of reductive algebraic groups

“…Building on this idea, for , a connected reductive linear algebraic group over a field , and , a subgroup of ( ), Serre introduced the concept of a ' -analogue' of semisimplification from representation theory in [19,Section 3.2.4]. This notion is also used for representations of various kinds of algebras: for example, see [12], [8], [16], [23], and [24]. It is also an ingredient in work of Lawrence-Sawin on the Shafarevich Conjecture for abelian varieties [13] and work of Lawrence-Venkatesh on Mordell's Conjecture [14], which involve Galois representations taking values in possibly non-connected reductive -adic groups.…”