Computing the associatied cycles of certain Harish-Chandra modules

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_$\mathbb{R}$)={\rm rank}(K_$\mathbb{R}$)$\mathrm{.}In\text{cite}MPVZweprovedthatforanyrepresentation$X$ofGelfand-Kirillovdimension$\frac{1}{2}\dim(G_{$\mathbb{R}$}/K_{$\mathbb{R}$})$, the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$ is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly.