Orthogonal roots and orbits of graded semisimple Lie algebras (Q1364260)

The theme of this paper is to give a structure of a class of regular prehomogeneous vector spaces such that they are obtained from the gradation of semisimple Lie algebras defined over a field of characteristic 0. They were originally studied by H. Rubenthaler and called prehomogeneous vector spaces of parabolic type. The target of the study of this paper is a prehomogeneous vector space of ``weakly'' commutative parabolic type, which is a generalization of the prehomogeneous vector space of commutative parabolic type and has a simple structure. One purpose of this paper is to give the orbit of ``1-simple'' elements of the prehomogeneous vector space such that there exist orthogonal roots whose sum of the co-roots is twice that of the element \(H_0\), where \(H_0\) is an element that characterizes the gradation of the Lie algebra.