Dirac cohomology and orthogonality relations for weight modules
type: Item
instance of: scholarly article
title: Dirac cohomology and orthogonality relations for weight modules
zbMATH Open document ID: 1504.17013
author: Yanyan Li
published in: Pacific Journal of Mathematics
publication date: 2022-09-12T00:00:00Z
review text: Let \(\mathfrak{g}\) be a complex reductive Lie algebra. Let \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\). Let \(\mathcal{M}(\mathfrak{g}, \mathfrak{h})\) be the category of finitely generated admissible \((\mathfrak{g}, \mathfrak{h})\) modules. An object in \(\mathcal{M}(\mathfrak{g}, \mathfrak{h})\) is called a weight module. Let \(M, N\) be two weight modules. Theorem 6.1 of the paper proves that the Euler-Poincare pairing between \(M\) and \(N\) equals the spinor pairing between them. This can be viewed as an analogue of Harish-Chandra's orthogonality relations of discrete series in the setting of weight modules. Theorem 7.1 further proves that the Euler-Poincare pairing between \(M\) and \(N\) equals the Dirac index pairing between them. Then the authors show in Theorem 7.9 that a simple weight module has non-zero Dirac cohomology if and only if it is a weight module with respect to a Borel subalgebra. As a consequence, Dirac cohomology of any simple weight module is determined.
reviewed by: Chao-Ping Dong
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 7583864
zbMATH Keywords: Dirac cohomology
zbMATH Keywords: weight module
zbMATH Keywords: Euler-Poincaré pairing
zbMATH Keywords: spinor pairing
MaRDI profile type: Publication
full work available at URL: pjm.2022.319.129
OpenAlex ID: W4293583447
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DOI: 10.2140/PJM.2022.319.129
description: scientific article; zbMATH DE number 7583864