nLab Dirac cohomology

Idea

Dirac cohomology appears in the effort to refine the nilpotent Lie algebra cohomology as it appears in the context of representation theory (in work of Bertram Kostant (Lie algebra cohomology and the generalized Borel-Weil theorem, Annals Math. 74 (1961) 329–387), related to Borel-Weil-Bott theorem).

References

• Jing-Song Huang, Pavle Pandžić?, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15:1 (2002) 185–202

• Bertram Kostant, Dirac cohomology for the cubic Dirac operator, in Studies in Memory of Issai Schur, Birkhauser, 2003; arXiv:math.RT/0208048; reprinted in collected works (pdf)

• Pavle Pandžić, Petr Somberg?, Higher Dirac cohomology of modules with generalized infinitesimal character, arxiv/1310.3570

• Jing-Song Huang, Pavle Pandžić, Dirac operators in representation theory, Representations of real and $p$-adic groups, 163–219, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 2, Singapore Univ. Press, 2004.

• J_S. Huang, Dirac cohomology, elliptic representations and endoscopy, in: Nevins, M., Trapa, P. (eds) Representations of Reductive Groups. Progress in Mathematics, vol 312. Birkhäuser 2015 doi

Woit wrote about relation to a case of Hilbert space from BRST cohomology:

• Peter Woit, Quantization and Dirac cohomology, pdf; BRST and Dirac cohomology, slides, pdf, BRST and Dirac cohomology, preprint pdf, 2009

• Spyridon Afentoulidis-Almpanis, Dirac cohomology for the BGG category $\mathcal{O}$, arXiv:2209.12566

category: algebra