Dirac Index and associated cycles of Harish-Chandra modules

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. For any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2} {\rm dim}(G_{\mathbb{R}}/K_{\mathbb{R}})$, we consider 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 $X$. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of $X$. This relationship was conjectured in \cite{MehdiPandzicVogan15}.