2018/11/17 by Wallach, Nolan R.
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.07195
The main result asserts: Let G be a reductive, affine algebraic group and let (ρ,V) be a regular representation of G. Let X be an irreducible ℂ × G invariant Zariski closed subset such that G has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, W,of X in the metric topology which is dense with complement of measure 0 such that if x ,y ∈ W then (ℂ × G )x is conjugate to (ℂ × G )y. Furthermore, if G x is a closed orbit of maximal dimension and if x is a smooth point of X then there exists y ∈ W such that (ℂ × G )x contains a conjugate of (ℂ × G )y. The proof involves using the Kempf-Ness theorem to reduce the result to the principal orbit type theorem for compact Lie groups.