The Bruhat–Chevalley Order of Parabolic Group Actions in General Linear Groups and Degeneration for Δ-Filtered Modules
✍ Scribed by T. Brüstle; L. Hille; G. Röhrle; G. Zwara
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 650 KB
- Volume
- 148
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
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