Actions of parabolic subgroups in GL_n on unipotent normal subgroups and quasi-hereditary algebras

Let R be a parabolic subgroup in $GL_n$. It acts on its unipotent radical $R_u$ and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra $k _t$ of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each bimodule B gives rise to an infinite number of those actions simultaneously: for each d in $ℕ^t$ we obtain a parabolic group R(d), which is the group of invertible elements in Λ(d), together with a unipotent normal subgroup U(d) in R(d). All those bimodules B are upper triangular with respect to the natural order of Λ. Then, according to [BH2], Theorem 1.1, there exists a quasi-hereditary algebra A such that the orbits of R(d) on U(d) are in bijection to the isomorphism classes of Δ-filtered A-modules of dimension vector d. We compute the quiver and relations of the quasi-hereditary algebra A corresponding to the action of the parabolic group R(d) on U(d). Moreover, we show that the Lie algebra of R(d) can be identified with the algebra Λ(d), and the Lie algebra of U(d) is isomorphic to a bimodule B(d) over Λ(d).