Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $$\mathcal{J}$\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\mathrm{Grk}(\mathcal{J})⩽\dim V\mathrm{/}{d}_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms.