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We study general properties of quantisations of Slodowy slices and discuss indetail the case of the minimal nilpotent orbit. Associated varieties of relatedprimitive ideals of U(g) are determined, in the general case, and the deVos-van Driel conjecture on Verma modules for finite W-algebras is proved inthe minimal nilpotent case.

Enveloping algebras of Slodowy slices and the Joseph ideal