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Dirac operators for Cartan motion groups

  • Spyridon Afentoulidis-Almpanis
  • Eyal Subag
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Abstract

We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.

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Cite

BibTeXRIS
BibTeX
@misc{afentoulidisalmpanis2026dirac,
  title = {{Dirac operators for Cartan motion groups}},
  author = {Spyridon Afentoulidis-Almpanis and Eyal Subag},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37245},
  eprint = {2609.37245},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37245},
}
RIS
TY  - UNPB
TI  - Dirac operators for Cartan motion groups
AU  - Spyridon Afentoulidis-Almpanis
AU  - Eyal Subag
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37245
UR  - https://arxiv.org/abs/2609.37245
AB  - We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.
ER  -