Dirac cohomology for the BGG category O

2022/09/26 by Spyridon Afentoulidis-Almpanis, Afentoulidis-Almpanis, Spyridon
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2209.12566

Abstract

We study Dirac cohomology H_D^\mathfrakg,\mathfrakh(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for H_D^\mathfrakg,\mathfrakh(M) while we show that in the case of a Hermitian symmetric pair (\mathfrakg,\mathfrakk) and an irreducible unitary module M\inO, 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 Pandžić and Somberg satisfy nice homological properties for M\inO.

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