  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
$\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, 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.
