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When a semisimple Lie group G acts on it's Lie algebra L, the semisimple elements of L and the nilpotent elements of L break up into a union of G-orbits. In the nilpotent case, it turns out that there are only finitely many orbits. Furthermore, there are known parameterizing sets for these orbits, known formulas to compute their dimensions, and a nice partial ordering on the set of orbits that relates the orbits to each other. We outline this classical theory, and in the case of the classical groups, show how two different parameterizing sets are related to each other.
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