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Additive relative invariants and the components of a linear free divisor

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Abstract

A prehomogeneous vector space is a rational representation $\rho:G\to GL(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. M. Sato showed that the hypersurface components of $D:=V\setminus \Omega$ are related to the characters $H\to GL(\mathbb{C})$ of $H$, an algebraic abelian quotient of $G$. Mimicking this work, we investigate the additive functions of $H$, homomorphisms $\Phi:H\to (\mathbb{C},+)$. Each such $\Phi$ is related to an `additive relative invariant', a rational function $h$ on $V$ such that $h\circ \rho(g)-h=\Phi(g)$ on $\Omega$ for all $g\in G$. 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 simplify proofs of existing results. We also examine the homotopy groups of $V\setminus D$.

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