Affine open orbits, reductive isotropy groups, and dominant gradient morphisms; a theorem of Mikio Sato

Frank Servedio

Vol. 72 (1977), No. 2, 537–545

DOI: 10.2140/pjm.1977.72.537

Abstract

An algebraic proof is given for a theorem of M. Sato. The theorem gives criteria for the open orbit in a prehomogeneous vector space under a reductive group to be an affine variety. The following conditions are equivalent:

1. O(G) the open orbit is an affine variety.
2. G_X the isotropy subgroup of X in O(G) is reductive.
3. There exists a semi-invariant form P of degree r ≧ 2 such that grad P : V → V ^∗ is a dominant morphism of affine varieties.

Mathematical Subject Classification 2000

Primary: 14L15

Secondary: 20G20

Milestones

Received: 4 November 1976

Revised: 23 March 1977

Published: 1 October 1977