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

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 G L(V). Now P acts on its unipotent radical P-u and on p(u) = Lie P-u, the Li e algebra of P-u, via the adjoint action. More generally, we consider the a ction of P on the lth member of the descending central series of p(u) denot ed by P-u((l)). All instances when P acts on p(u)((l)) for l greater than o r equal to 0 with a finite number of orbits are known. In this note, we giv e a complete combinatorial description of the closure relation on the set o f P-orbits on p(u)((l)), i.e., the Bruhat-Chevalley order, for every finite case. There is a canonical bijection between the set of P-orbits on p(u)(( l)) and the set of isomorphism classes of Delta-filtered modules of a parti cular dimension vector e of a certain quasi-hereditary algebra A(t, l). The se isomorphism classes in turn are given by the orbits of a reductive group G(e) on the variety R(Delta)(e) of all A(t, l)-modules with Delta-filtrati on and dimension vector e. The subcategory of A(t, l)-mod of all Delta-filt ered A(t, l)-modules of dimension vector e is denoted by F(Delta)(e). In ou r chief res-nit, Theorem 1.1, we show that provided there is only a finite number of isomorphism classes of indecomposable modules in F(Delta) the fol lowing three posets coincide: (1) the Bruthat-Chevalley order on the set of P-orbits on p(u)((l)); (2) the Bruhat-Chevalley order on the set of G(e)-orbits on R(Delta)(e); (3) the poset opposite to the so called hom-order on the set of isomorphism classes of F(Delta)(e). The advantage of this hom-order is that it is given purely by discrete inva riants and that it can be computed explicitly for any given finite case. Th eorem 1.1 then in turn allows us to explicitly determine the closure relati ons for the P-orbits on p(u)((l)) with the aid of this hem-order. We presen t some examples of Hasse diagrams in an Appendix. (C) 1999 Academic Press.