Science China Mathematics

, Volume 62, Issue 11, pp 2287–2308 | Cite as

Low degree cohomologies of congruence groups

  • Jian-Shu Li
  • Binyong SunEmail author


We prove the vanishing of certain low degree cohomologies of some induced representations. As an application, we determine certain low degree cohomologies of congruence groups.


cohomology congruence group automorphic form 


22E41 22E47 



The first author was supported by Research Grants Council General Research Fund of Hong Kong Special Administrative Region (Grant No. 16303314). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11688101, 11525105, 11621061 and 11531008).


  1. 1.
    Bernstein J, Krötz B. Smooth Frechet globalizations of Harish-Chandra modules. Israel J Math, 2014, 199: 45–111MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blanc P. Sur la cohomologie continue des groupes localement compacts. Ann Sci École Norm Sup (4), 1979, 12: 137–168MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borel A. Stable real cohomology of arithmetic groups. Ann Sci École Norm Sup (4), 1974, 7: 235–272MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borel A, Wallach N. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed. Mathematical Surveys and Monographs, 67. Providence: American Mathematical Society, 2000CrossRefGoogle Scholar
  5. 5.
    Casselman W. Canonical extensions of Harish-Chandra moudles to representations of G. Canada J Math, 1989, 41: 385–438MathSciNetCrossRefGoogle Scholar
  6. 6.
    Enright T J. Reductive Lie algebra cohomology and unitary representations of complex Lie groups. Duke Math J, 1979, 46: 513–525MathSciNetCrossRefGoogle Scholar
  7. 7.
    Franke J. Harmonic analysis in weighted L 2-spaces. Ann Sci École Norm Sup (4), 1998, 31: 181–279MathSciNetCrossRefGoogle Scholar
  8. 8.
    Garland H. A finiteness theorem for K 2 of a number field. Ann of Math (2), 1971, 94: 534–548MathSciNetCrossRefGoogle Scholar
  9. 9.
    Garland H, Hsiang W C. A square integrability criterion for the cohomology of arithmetic groups. Proc Nat Acad Sci USA, 1968, 59: 354–360MathSciNetCrossRefGoogle Scholar
  10. 10.
    Harish-Chandra. Automorphic Forms on Semisimple Lie Groups. Lecture Notes in Mathematics, vol. 62. Berlin-Heidelberg-New York: Springer, 1968CrossRefGoogle Scholar
  11. 11.
    Hochschild G, Mostow G. Cohomology of Lie groups. Illinois J Math, 1962, 6: 367–401MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kaneyuki S, Nagano T. On certain quadratic forms related to symmetric riemannian spaces. Osaka Math J, 1962, 14: 241–252MathSciNetzbMATHGoogle Scholar
  13. 13.
    Knapp A. Lie Groups Beyond an Introduction, 2nd ed. Progress in Mathematics, vol. 140. Boston: Birkhäuser, 1996CrossRefGoogle Scholar
  14. 14.
    Knapp A, Vogan D. Cohomological Induction and Unitary Representations. Princeton: Princeton University Press, 1995Google Scholar
  15. 15.
    Kumaresan S. On the canonical k-types in the irreducible unitary g-modules with nonzero relative cohomology. Invent Math, 1980, 59: 1–11Google Scholar
  16. 16.
    Langlands R P. On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Mathematics, vol. 544. Berlin-New York: Springer-Verlag, 1976CrossRefGoogle Scholar
  17. 17.
    Li J-S, Schwermer J. On the Eisenstein cohomology of arithmetic groups. Duke Math J, 2004, 123: 141–169MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li J-S, Schwermer J. Automorphic representations and cohomology of arithmetic groups. In: Challenges for the 21st Century. Proceedings of the International Conference on Fundamental Sciences. Singapore: World Scientific, 2001, 102–137CrossRefGoogle Scholar
  19. 19.
    Moeglin C, Waldspurger J L. Spectral Decomposition and Eisenstein Series. Cambridge Tracts in Mathematics, vol. 113. Cambridge: Cambridge University Press, 1995CrossRefGoogle Scholar
  20. 20.
    Schwermer J. Geometric cycles, arithmetic groups and their cohomology. Bull Amer Math Soc, 2010, 47: 187–279MathSciNetCrossRefGoogle Scholar
  21. 21.
    Vogan D. Unitarizability of certain series of representations. Ann of Math (2), 1984, 120: 141–187MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vogan D, Zuckerman G. Unitary representations with non-zero cohomology. Compos Math, 1984, 53: 51–90zbMATHGoogle Scholar
  23. 23.
    Wallach N. Real Reductive Groups I. San Diego: Academic Press, 1988Google Scholar
  24. 24.
    Wallach N. Real Reductive Groups II. San Diego: Academic Press, 1992zbMATHGoogle Scholar
  25. 25.
    Wallach N. On the unitarizability of derived functor modules. Invent Math, 1984, 78: 131–141CrossRefGoogle Scholar
  26. 26.
    Yang J. On the real cohomology of arithmetic groups and the rank conjecture for number fields. Ann Sci École Norm Sup (4), 1992, 25: 287–306MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Advanced Study in MathematicsZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsHong Kong University of Science and TechnologyHong KongChina
  3. 3.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  4. 4.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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