Integral invariant functions on the nilpotent elements of a semisimple Lie algebra

Abstract:

Let L be a semisimple Lie algebra over an algebraically closed field of characteristic zero. It is shown that there is a finitely generated ring R of integral invariant functions such that for nilpotent elements x and y of L, one has x conjugate to y if and only if $f(x) = f(y)$ for all f in R. The result is analogous to Chevalley’s determination of conjugacy classes of semisimple elements by the ring of invariant polynomial functions.