Computing the associatied cycles of certain Harish-Chandra modules

TL;DR

This paper explicitly computes the coefficients relating the Dirac index polynomial to the multiplicities in the associated cycle for certain Harish-Chandra modules of real Lie groups.

Contribution

It provides an explicit calculation of the coefficients connecting Dirac index polynomials and associated cycle multiplicities for specific Harish-Chandra modules.

Findings

Explicit formulas for the coefficients are derived.

The results clarify the relationship between Dirac index and associated cycles.

Applications to representation theory of real Lie groups are discussed.

Abstract

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