Additive relative invariants and the components of a linear free divisor

الأصل · EN

A 'prehomogeneous vector space' is a rational representation ρ:G(V) of a connected complex linear algebraic group G that has a Zariski open orbit Ω⊂ V. Mikio Sato showed that the hypersurface components of D:=V Ω are related to the rational characters H(C) of H, an algebraic abelian quotient of G. Mimicking this work, we investigate the 'additive functions' of H, the 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 helps describe 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 simplify proofs of existing results. We also examine the homotopy groups of V D.