A characterization of finite prehomogeneous vector spaces of $D_4$-type under various scalar restrictions

Abstract

In the present paper, we give conditions to have only finitely many orbits for prehomogeneous vector spaces of $D_4$-type. This paper completes the classification of finite prehomogeneous vector spaces of type $(G \times SL_n, \rho \otimes \Lambda_1)$ with $n \geq 2$. We consider everything over the complex number field $\mathbb{C}$.