A note on Verma modules for finite W-algebras

A finite W-algebra U(g,e) is a certain finitely generated algebra associated to a nilpotent element e of a complex reductive Lie algebra g. There is a (loop) filtration on U(g,e) such that the associated graded algebra is isomorphic to U(ge), where ge is the centralizer of e in g. In this short note, we show that Verma modules for finite W-algebras, as defined in Brundan et al. (2008) [BGK], are filtered deformations of Verma modules for U(ge).