Dirac Index and associated cycles of Harish-Chandra modules

Abstract

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that $\mathrm{rank}(G_{\mathbb{R}})=\mathrm{rank}(K_{\mathbb{R}})$. For any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2}\dim (G_{\mathbb{R}}/K_{\mathbb{R}})$, 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}.