Commutative quotients of finite W-algebras

In this paper we reduce the problem of 1-dimensional representations for the finite W-algebras and Humphreys' conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional Lie algebras. We use Katsylo's results on sections of sheets to determine the Krull dimension of the largest commutative quotient of the finite W-algebra U(g,e).