Application II: Periods, Distinguished Representations and \((\mathfrak {g},K)\)-cohomologies

  • Toshiyuki Kobayashi
  • Birgit Speh
Part of the Lecture Notes in Mathematics book series (LNM, volume 2234)


Let H be a subgroup of G. Following the terminology used in automorphic forms and the relative trace formula, we say that a smooth representation U of G is H-distinguished if there exists a nontrivial H-invariant linear functional.


  1. 5.
    N. Bergeron, L. Clozel, Spectre automorphe des variétés hyperboliques et applications topologiques, in Astérisque, vol. 303 (Soc. Math. France, 2005), xx+218 pp.Google Scholar
  2. 6.
    N. Bergeron, J. Millson, C. Mœglin, Hodge type theorems for arithmetic manifolds associated to orthogonal groups. Int. Math. Res. Not. IMRN 2017(15), 4495–4624MathSciNetzbMATHGoogle Scholar
  3. 9.
    A. Borel, N.R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups: Second edition. Math. Surveys Monogr., vol. 67 (Amer. Math. Soc., Providence, RI, 2000), xviii+260 pp.; First ed., Ann. of Math. Stud., vol. 94 (Princeton Univ. Press, Princeton, NJ, 1980), xvii+388 pp.Google Scholar
  4. 12.
    W.T. Gan, B.H. Gross, D. Prasad, J.-L. Waldspurger, Sur les conjectures de Gross et Prasad. I, in Astérique, vol. 346 (Soc. Math. France, 2012), xi+318 pp.Google Scholar
  5. 18.
    A. Ichino, T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture. Geom. Funct. Anal. 19, 1378–1425 (2010)MathSciNetCrossRefGoogle Scholar
  6. 19.
    A. Ichino, S. Yamana, Periods of automorphic forms: the case of (GLn+1 ×GLn, GLn). Compos. Math. 151, 665–712 (2015)MathSciNetCrossRefGoogle Scholar
  7. 24.
    A.W. Knapp, D.A. Vogan, Jr., Cohomological Induction and Unitary Representations. Princeton Math. Ser., vol. 45 (Princeton Univ. Press, Princeton, NJ, 1995), xx+948 pp. ISBN: 978-0-691-03756-6Google Scholar
  8. 46.
    S.S. Kudla, J.J. Millson, Geodesic cyclics and the Weil representation. I. Quotients of hyperbolic space and Siegel modular forms. Compos. Math. 45, 207–271 (1982)zbMATHGoogle Scholar
  9. 48.
    S.S. Kudla, J.J. Millson, The theta correspondence and harmonic forms. II. Math. Ann. 277, 267–314 (1987)MathSciNetCrossRefGoogle Scholar
  10. 54.
    B. Sun, The nonvanishing hypothesis at infinity for Rankin–Selberg convolutions. J. Am. Math. Soc. 30, 1–25 (2017)MathSciNetCrossRefGoogle Scholar
  11. 56.
    Y.L. Tong, S.P. Wang, Geometric realization of discrete series for semisimple symmetric spaces. Invent. Math. 96, 425–458 (1989)MathSciNetCrossRefGoogle Scholar
  12. 58.
    J.A. Vargas, Restriction of some discrete series representations. Algebras Groups Geom. 18, 85–99 (2001)MathSciNetzbMATHGoogle Scholar
  13. 61.
    D.A. Vogan, Jr., G.J. Zuckerman, Unitary representations with nonzero cohomology. Compos. Math. 53, 51–90 (1984)MathSciNetzbMATHGoogle Scholar
  14. 63.
    S.P. Wang, Correspondence of modular forms to cycles associated to O(p, q). J. Differ. Geom. 22, 151–213 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Toshiyuki Kobayashi
    • 1
    • 2
  • Birgit Speh
    • 3
  1. 1.Graduate School of Mathematical SciencesThe University of TokyoKomabaJapan
  2. 2.Kavli IPMUKashiwaJapan
  3. 3.Department of MathematicsCornell UniversityIthacaUSA

Personalised recommendations