On the separation property of orbits in representation spaces

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 (α,β) of linearly independent linear forms on V there is a point x on X such that α(x)=0 and β(x)≠0. A more geometric way to express this is the following: every linear subspace H V of codimension 1 is linearly spanned by its intersection with X. The separation property was first asked for conjugacy classes in simple Lie algebras, in connection with some classification problems. We give a general answer for orbits in representation spaces of algebraic groups and discuss in detail some special cases. We also introduce a strong and a weak separation property which come up very naturally in our setting. It turns out that these separation properties have a number of very nice features. For example, we discovered the surprising fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit, and we show that a generic orbit always has the separation property.