Dirac cohomology for the BGG category $\mathcal{O}$

TL;DR

This paper proves Vogan's conjecture for Dirac cohomology in category O of semisimple Lie algebras, showing nonvanishing results and relations to nilpotent cohomology, with implications for higher Dirac cohomology and index theory.

Contribution

It establishes Vogan's conjecture for modules in category O and relates Dirac cohomology to nilpotent Lie algebra cohomology in Hermitian symmetric cases.

Findings

Proved nonvanishing of Dirac cohomology for category O modules.

Showed Dirac cohomology coincides with nilpotent cohomology in specific cases.

Demonstrated homological properties of higher Dirac cohomology and index.

Abstract

We study Dirac cohomology $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ for modules belonging to category $\mathcal{O}$ of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ while we show that in the case of a Hermitian symmetric pair $(\mathfrak{g},\mathfrak{k})$ and an irreducible unitary module $M\in\mathcal{O}$, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in $M$. In the last part, we show that the higher Dirac cohomology and index introduced by Pand\v{z}i\'c and Somberg satisfy nice homological properties for $M\in\mathcal{O}$.