Formes réelles des espaces préhomogènes irréductibles de type parabolique. (Real forms of irreducible prehomogeneous vector spaces of parabolic type)
The main result of this paper is the determination of all real forms of a regular prehomogeneous vector space of parabolic type. Let \(V_{{\mathbb{C}}}\) be a complex finite dimensional vector space. For a complex linear algebraic subgroup \(G_{{\mathbb{C}}}\) in \(GL(V_{{\mathbb{C}}})\) we call the pair \((G_{{\mathbb{C}}},V_{{\mathbb{C}}})\) a prehomogeneous vector space if there exists a Zariski-open \(G_{{\mathbb{C}}}\)-orbit in \(V_{{\mathbb{C}}}\). The author classified all the prehomogeneous vector spaces appearing in simple complex Lie algebras and called them prehomogeneous vector spaces of commutative parabolic type in his previous work. In this paper, he has determined all the real forms of such prehomogeneous vector spaces by calculating possible real forms of the root systems. His list covers all the regular irreducible prehomogeneous vector spaces. Real forms are important when we consider the local zeta function of regular prehomogeneous vector spaces.
regular prehomogeneous vector space of parabolic type
scientific article; zbMATH DE number 3944092