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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prehomogeneous vector spaces and varieties

Author: Frank J. Servedio
Journal: Trans. Amer. Math. Soc. 176 (1973), 421-444
MSC: Primary 20G05
MathSciNet review: 0320173
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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.

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Keywords: Linear algebraic group, affine variety, Zariski open orbit, semi-invariant form, reductive linear algebraic group, Borel subgroup, simple algebraic group, irreducible representation
Article copyright: © Copyright 1973 American Mathematical Society