Hyperplane sections and separation of orbits in representation spaces

Hanspeter Kraft (Basel)

A subset X of a complex 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 linear forms on V there is a point x on X such that a(x) = 0 and b(x) is non-zero. A more geometric way to express this is the following: Every linear subspace H of V of codimension 1 is linearly spanned by its intersection with X. The question of existence of such subsets was asked by J.C. Jantzen in connection with a classification problem of Lie algebras and with orbits in representation spaces of algebraic groups. It turned out to be a very interesting property having a number of nice features. For example, we discovered the surprising fact that in an irreducible representation of a connected semisimple group every hyperplane meets every orbit. We can classify the representions where all orbits have the separation property. Also we can show that a generic orbit always has the separation property.

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