Rational Covariants of Reductive Groups and Homaloidal Polynomials

Let $G$ be a complex reductive group, $V$ a $G$-module and $f\in\OOO(V)^G$ a nonconstant homogeneous invariant. We investigate relations between the following properties: \roster \bullitem $df\colon V\to V^*$ is dominant, \bullitem $f$ is {\it homaloidal}, i.e., $df$ induces a birational map $\PP(V)\to \PP(V^*)$, \bullitem $V$ is stable, i.e., the generic $G$-orbit is closed. \endroster If $f$ generates ${\Cal O}(V)^G$, we show that the properties are equivalent, generalizing results of \name{Sato-Kimura} on prehomogeneous vector spaces.

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Published 1 January 2001