Dirac index of some unitary representations of $Sp(2n, \mathbb{R})$ and $SO^*(2n)$

TL;DR

This paper computes the Dirac index for a broad class of unitary representations of $Sp(2n, R)$ and $SO^*(2n)$, revealing insights into their structure and proposing a conjecture about their Dirac cohomology.

Contribution

It provides explicit calculations of the Dirac index for many unitary representations of these groups and introduces a conjecture linking all such representations with nonzero Dirac cohomology.

Findings

Computed Dirac indices for weakly fair $A_q(\\lambda)$ modules and unipotent representations.

Established a connection between Dirac index cancellations and spin-lowest $K$-types.

Conjectured that all unitary representations with nonzero Dirac cohomology are included in this class.

Abstract

Let $G$ be $Sp(2n, \mathbb{R})$ or $SO^*(2n)$. We compute the Dirac index of a large class of unitary representations considered by Vogan in Section 8 of [Vog84], which include all weakly fair $A_{\mathfrak{q}}(\lambda)$ modules and (weakly) unipotent representations of $G$ as two extreme cases. We conjecture that these representations exhaust all unitary representations of $G$ with nonzero Dirac cohomology. In general, for certain irreducible unitary module of an equal rank group, we clarify the link between the possible cancellations in its Dirac index, and the parities of its spin-lowest $K$-types.