Prehomogeneous Spaces for Parabolic Group Actions in Reductive Algebraic Groups

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Actions of algebraic groups on vector spaces and algebraic varieties play an important role in many branches of pure and applied mathematics. Among these groups actions, the actions of reductive groups are the best understood. The next natural class is formed by parabolic subgroups of reductive groups. The action of a parabolic group on the Lie algebra of its unipotent radical is in some sense as fundamental as the adjoint action of a reductive group. We intend to study particular geometric questions about this action.More specifically, for a given parabolic subgroup P of a reductive group we intend to classify all P-submodules of the nilradical of Lie P which admit a dense P-orbit. Our strategies for obtaining solutions of this problem naturally vary depending on the type of the ambient reductive group.For general linear groups we hope to employ a representation theoretic reformulation of this problem in terms of the existence of particular types of modules for certain quasi-hereditary algebras.Different techniques should then allow us to deduce the solution of our problem for the other classical groups from the solution for the general linear groups.We hope to address this problem for the groups of exceptional type by computational means.Finally, we plan to complete the classification of all finite orbit modules among the members of the descending central series of the nilradicals of all parabolic subgroups of simple algebraic groups by studying the open cases for groups of type E7 and E8.