  A highest weight theory for a finite W-algebra U(g,e) was developed in [BGK].
This leads to a strategy for classifying the irreducible finite dimensional
U(g,e)-modules. The highest weight theory depends on the choice of a parabolic
subalgebra of g leading to different parameterizations of the finite
dimensional irreducible U(g,e)-modules. We explain how to construct an
isomorphism preserving bijection between the parameterizing sets for different
choices of parabolic subalgebra when g is of type A, or when g is of types C or
D and e is an even multiplicity nilpotent element
