An Explicit Formula for Szegő Kernels on the Heisenberg Group


In this paper, we give an explicit formula for the Szegő kernel for (0, q) forms on the Heisenberg group Hn+1.

This is a preview of subscription content, access via your institution.


  1. [1]

    Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Astérisque, 34–35, 123–164 (1976)

    MATH  Google Scholar 

  2. [2]

    Cheng, J. H., Malchiodi, A., Yang, P.: A positive mass theorem in three dimensional Cauchy–Riemann geometry. Adv. Math., 308, 276–347 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    Chen, S. C., Shaw, M. C.: Partial Differential Equations in Several Complex Variables, AMS/IP Studies in Advanced Mathematics, 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001

    Google Scholar 

  4. [4]

    Grigis, A., Sjöstrand, J.: Microlocal Analysis for Differential Operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994

    Google Scholar 

  5. [5]

    Hsiao, C. Y.: Projections in several complex variables. Mém. Soc. Math. France, Nouv. Sér., 123, 131 (2010)

    MathSciNet  MATH  Google Scholar 

  6. [6]

    Hsiao, C. Y., Marinescu, G.: On the singularities of the Szegő projections on lower energy forms. J. Differential Geom., 107(1), 83–155 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    Hsiao, C. Y., Marinescu, G.: Szegő kernel asymptotics and morse inequalties on CR manifolds. Math. Z., 271, 509–553 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    Hsiao, C. Y., Yung, P. Y.: The tangential Cauchy–Riemann complex on the heisenberg group via conformal invariance. Bull. Inst. Math. Acad. Sin. (N.S.), 8(3), 359–375 (2013)

    MathSciNet  MATH  Google Scholar 

  9. [9]

    Hsiao, C. Y., Yung, P. Y.: Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3. Adv. Math., 281, 734–822 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    Hua, L. K.: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Transl. of Math. Monographs 6, American Math. Society, 1963

    Google Scholar 

  11. [11]

    Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Math., 254, Birkhäuser Verlag, Basel, 2007

    Google Scholar 

  12. [12]

    Szegő, G.: über orthogonalsysteme von polynomen. Math. Z., 4, 139–151 (1919)

    MathSciNet  Article  MATH  Google Scholar 

Download references


The authors would like to thank the Institute for Mathematics, National University of Singapore for hospitality, a comfortable accommodation and financial support during their visits in May, 2017 for the program “Complex Geometry, Dynamical Systems and Foliation Theory”. A main part of this work was done when the first and third author were visiting the Institute of Mathematics, Academia Sinica in January, 2017. The authors thank the referees for carefully reading the manuscript and giving useful advices which improve the presentation of this paper.

Author information



Corresponding author

Correspondence to Hendrik Herrmann.

Additional information

In Memory of Professor Qikeng Lu (1927–2015)

The first author was partially supported by the CRC TRR 191: “Symplectic Structures in Geometry, Algebra and Dynamics”; the second author was partially supported by Taiwan Ministry of Science of Technology project (Grant No. 104-2628-M-001-003-MY2) and the Golden-Jade fellowship of Kenda Foundation; the third author was supported by National Natural Science Foundation of China (Grant No. 11501422)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Herrmann, H., Hsiao, C.Y. & Li, X.S. An Explicit Formula for Szegő Kernels on the Heisenberg Group. Acta. Math. Sin.-English Ser. 34, 1225–1247 (2018).

Download citation


  • Heisenberg group
  • Szegő kernels
  • complex Fourier integral operators

MR(2010) Subject Classification

  • 32V20
  • 32A25
  • 32W10