The separation properties for closures of toric orbits

Abstract

A subset X of a vector space V is said to have the "Separation Property" if it separates linear forms in the following sense: given a pair (a, b) of linearly independent forms on V there is a point x on X such that a(x)=0 and b(x) is not equal to 0. A more geometric way to express this is the following: every homogeneous hyperplane H in V is linearly spanned by its intersection with X. The separation property was first asked for conjugacy classes in simple Lie algebras. We give an answer for orbit closures in representation spaces of an algebraic torus. We consider also the strong and the weak separation properties. It turns out that toric orbits well illustrate these concepts.