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

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 $A_{\mathfrak{q}}(λ)$ 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.

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