ADDITIVE RELATIVE INVARIANTS AND THE COMPONENTS OF A LINEAR FREE DIVISOR

Abstract. A prehomogeneous vector space is a rational representation ρ: G → GL(V) of a connected complex linear algebraic group G that has a Zariski open orbit Ω ⊂ V. M. Sato showed that the hypersurface components of D: = V \Ω are related to the characters H → GL(C) of H, an algebraic abelian quotient of G. Mimicking this work, we investigate the additive functions of H, homomorphisms Φ: H → (C,+). Each such Φ is related to an additive relative invariant, a rational function h on V such that h ◦ ρ(g) − h = Φ(g) on Ω for all g ∈ G. Such an h is homogeneous of degree 0, and describes the behavior of certain subsets of D under the G–action. For those prehomogeneous vector spaces with D a type of hypersurface called a linear free divisor, we prove there are no nontrivial additive functions of H, and hence H is an algebraic torus. From this we gain insight into the structure of such representations and prove that the number of irreducible components of D equals the dimension of the abelianization of G. For some special cases (G abelian, reductive, or solvable, or D irreducible) we simplif