On changing highest weight theories for finite W-algebras

A highest weight theory for a finite W-algebra $U(\mathfrak{g},e)$ was developed in Brundan et al. (Int. Math. Res. Not. 15:rnn051, 2008). This leads to a strategy for classifying the irreducible finite dimensional $U(\mathfrak{g},e)$ -modules. The highest weight theory depends on the choice of a parabolic subalgebra of $\mathfrak{g}$ leading to different parameterizations of the finite dimensional irreducible $U(\mathfrak{g},e)$ -modules. We explain how to construct an isomorphism preserving bijection between the parameterizing sets for different choices of parabolic subalgebra when $\mathfrak{g}$ is of type A, or when $\mathfrak{g}$ is of types C or D and e is an even multiplicity nilpotent element.