Highest Weight Theory for Finite W-Algebras

We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step toward the classification of finite-dimensional irreducible modules. We also introduce an analogue of the BGG category
$\mathcal O$
. Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.