Dirac Index and associated cycles of Harish-Chandra modules

TL;DR

This paper explores the relationship between Dirac index polynomials and associated cycles of certain Harish-Chandra modules in the context of equal rank real Lie groups, confirming a conjecture under specific conditions.

Contribution

It establishes an explicit link between Dirac index polynomials and associated cycle multiplicities for Harish-Chandra modules, confirming a prior conjecture under a Springer correspondence condition.

Findings

Derived an explicit formula relating Dirac index polynomials to associated cycle multiplicities.

Confirmed the conjecture from Mehdi Pandzic and Vogan (2015) under a Springer correspondence condition.

Provided a new tool for understanding the structure of Harish-Chandra modules in representation theory.

Abstract

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}.