On a Class of Multiplicity-Free Nilpotent KC-Orbits

An action of an algebraic reductive groupG on an affine varietyM is calledmultiplicity-free if the multiplicity of any particular irreducible representation of G in the space C [M ] of regular functions on M is at most one: In [1], Kac provides a complete list of multiplicity-free actions for the case when G is a connected reductive algebraic group and M is a finite-dimensional vector space upon which G acts by an irreducible representation. Kac studied this case primarily to get an accounting of the possibilities for the action of [g0, g0] on gi, where g = ∑ i∈Z gi is a Z-graded semisimple Lie algebra over C.