Prehomogeneous Vector Spaces and Varieties

Abstract

An affine algebraic group G over an algebraically closed field k of characteristic 0 is said to act prehomogeneously on an affine variety W over k if G has a (unique) open orbit $o(G)$ in W. When W is the variety of points of a vector space V, $G \subseteq GL(V)$ and G acts prehomogeneously and irreducibly on V (We say an irreducibly prehomogeneous pair (G, V).), the following conditions are shown to be equivalent: 1. the existence of a nonconstant semi-invariant P in $k[V] \cong S({V^\ast })$, 2. $(G’,V)$ is not a prehomogeneous pair ($G’$ is the commutator subgroup of G, a semisimple closed subgroup of G.), 3. if $X \in o(G)$, then $G_X^0 \subseteq G’$. ($G_X^0$ is the connected identity component of ${G_X}$, the stabilizer of X in G.) Further, if such a P exists, the criterion, due to Mikio Sato, “$o(G)$ is the principal open affine ${U_P}$ if and only if $G_X^0$ is reductive” is stated. Under the hypothesis G reductive, the condition “there exists a Borel subgroup $B \subseteq G$ acting prehomogeneously on W” is shown to be sufficient for $G\backslash W$, the set of G-orbits in the affine variety W to be finite. These criteria are then applied to a class of irreducible prehomogeneous pairs (G, V) for which $G’$ is simple and three further conjectures, one due to Mikio Sato, are stated.