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}$)$\mathrm{.}Foranyrepresentation$X$ofGelfand-Kirillovdimension$\frac{1}{2} {\rm dim}(G_{$\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}.