Home Dirac cohomology and geometric quantization
Article
Licensed
Unlicensed Requires Authentication

Dirac cohomology and geometric quantization

  • Meng-Kiat Chuah EMAIL logo and Jing-Song Huang
Published/Copyright: July 9, 2014

Abstract

Let G be a connected real semisimple Lie group having a finite center and a compact Cartan subgroup T with Lie algebra 𝔱0. Let ω be a G×T-invariant symplectic form on X=G×𝔱0. We incorporate Dirac cohomology into the geometric quantization of (X,ω) and study the resulting multiplicity-free unitary G×T-representation on a Hilbert space (X,ω). We also perform symplectic reduction of (X,ω) and show that our quantization method satisfies the principle “quantization commutes with reduction”. As an application we construct various models of discrete series.

Funding statement: The first named author is supported by a research grant from the Ministry of Science and Technology of Taiwan. The second named author is supported by research grants from the Research Grant Council of Hong Kong SAR and the National Science Foundation of China.

Acknowledgements

The authors are very grateful to the referees for their detailed comments and kind help.

References

[1] Abraham R. and Marsden J., Foundations of mechanics, 2nd ed., Addison-Wesley, Redwood City 1985. Search in Google Scholar

[2] Atiyah M. and Schmid W., A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. 10.1007/978-94-009-8961-0_7Search in Google Scholar

[3] Atiyah M. and Schmid W., Erratum: A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 54 (1979), 189–192. 10.1007/978-94-009-8961-0_8Search in Google Scholar

[4] Chuah M. K., Kaehler structures on K/(P,P), Trans. Amer. Math. Soc. 349 (1997), 3373–3390. 10.1090/S0002-9947-97-01840-0Search in Google Scholar

[5] Chuah M. K., Holomorphic discrete models of semisimple Lie groups and their symplectic constructions, J. Funct. Anal. 175 (2000), 17–51. 10.1006/jfan.2000.3586Search in Google Scholar

[6] Chuah M. K., Regular principal models of split semisimple Lie groups, J. reine angew. Math. 623 (2008), 195–211. 10.1515/CRELLE.2008.076Search in Google Scholar

[7] Chuah M. K. and Guillemin V., Kaehler structures on K/N, The Penrose transform and analytic cohomology in representation theory (South Hadley 1992), Contemp. Math. 154, American Mathematical Society, Providence (1993), 181–195. 10.1090/conm/154/01363Search in Google Scholar

[8] Dong C.-P. and Huang J.-S., Jacquet modules and Dirac cohomology, Adv. Math. 226 (2011), 2911–2934. 10.1016/j.aim.2010.09.024Search in Google Scholar

[9] Gelfand I. M. and Zelevinskii A., Models of representations of classical groups and their hidden symmetries, Funct. Anal. Appl. 18 (1984), 183–198. 10.1007/BF01086156Search in Google Scholar

[10] Guillemin V. and Sternberg S., Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515–538. 10.1007/BF01398934Search in Google Scholar

[11] Guillemin V. and Sternberg S., Symplectic techniques in physics, Cambridge University Press, Cambridge 1984. Search in Google Scholar

[12] Harish-Chandra , Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. 10.1007/BF02392813Search in Google Scholar

[13] Huang J.-S., Dirac cohomology and Dirac induction, Sci. China Math. 54 (2011), 2373–2381. 10.1007/s11425-011-4295-5Search in Google Scholar

[14] Huang J.-S., Kang Y.-F. and Pandžić P., Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009), 163–173. 10.1007/s00031-008-9036-7Search in Google Scholar

[15] Huang J.-S. and Pandžić P., Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185–202. 10.1090/S0894-0347-01-00383-6Search in Google Scholar

[16] Huang J.-S. and Pandžić P., Dirac operators in representation theory, Birkhäuser-Verlag, Basel 2006. Search in Google Scholar

[17] Huang J.-S., Pandžić P. and Protsak V., Dirac cohomology of Wallach representations, Pacific J. Math. 250 (2011), 163–190. 10.2140/pjm.2011.250.163Search in Google Scholar

[18] Huang J.-S., Pandžić P. and Renard D., Dirac operators and Lie algebra cohomology, Represent. Theory 10 (2006), 299–313. 10.1090/S1088-4165-06-00267-6Search in Google Scholar

[19] Huang J.-S., Pandžić P. and Zhu F.-H., Dirac cohomology, K-characters and branching laws, Amer. J. Math. 135 (2013), 1253–1269. 10.1353/ajm.2013.0041Search in Google Scholar

[20] Huang J.-S. and Xiao W., Dirac cohomology of highest weight modules, Selecta Math. (N.S.) 18 (2012), 803–824. 10.1007/s00029-011-0085-8Search in Google Scholar

[21] Kostant B., Quantization and unitary representations, Lectures in modern analysis and applications III, Lecture Notes in Math. 170, Springer-Verlag, Berlin (1970), 87–208. 10.1007/BFb0079068Search in Google Scholar

[22] Kostant B., A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), 447–501. 10.1215/S0012-7094-99-10016-0Search in Google Scholar

[23] Kostant B., Dirac cohomology for the cubic Dirac operator, Studies in memory of Issai Schur, Progr. Math. 210, Birkhäuser-Verlag, Basel (2003), 69–93. 10.1007/978-1-4612-0045-1_4Search in Google Scholar

[24] Marsden J. and Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121–130. 10.1016/0034-4877(74)90021-4Search in Google Scholar

[25] Meise R. and Vogt D., Introduction to functional analysis, Oxf. Grad. Texts Math. 2, Clarendon Press, Oxford 1997. 10.1093/oso/9780198514855.001.0001Search in Google Scholar

[26] Parthasarathy R., Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. 10.2307/1970892Search in Google Scholar

[27] Schmid W., L2-cohomology and the discrete series, Ann. of Math. (2) 103 (1976), 375–394. 10.2307/1970944Search in Google Scholar

[28] Vogan, Jr. D. A., Dirac operators and unitary representations, three talks at MIT Lie groups seminar, Fall 1997. 10.1016/B978-012625440-2/50006-9Search in Google Scholar

[29] Vogan, Jr. D. A., The method of coadjoint orbits for real reductive groups, Representation theory of Lie groups (Park City 1998), IAS/Park City Math. Ser. 8, American Mathematical Society, Providence (2000), 177–238. 10.1090/pcms/008/05Search in Google Scholar

Received: 2012-12-3
Revised: 2014-4-18
Published Online: 2014-7-9
Published in Print: 2016-11-1

© 2016 by De Gruyter

Downloaded on 14.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2014-0050/html?lang=en
Scroll to top button