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.
finite w-algebras representation theory