Transformation Groups

, Volume 14, Issue 1, pp 41–86 | Cite as

Restriction De La Cohomologie D’une Variété De Shimura à Ses Sous-variétés

Article

Abstract

Let G be a connected semisimple group over \( \mathbb{Q} \). Given a maximal compact subgroup KG(\( \mathbb{R} \)) such that X = G(\( \mathbb{R} \))/K is a Hermitian symmetric domain, and a convenient arithmetic subgroup Γ ⊂ G(\( \mathbb{Q} \)), one constructs a (connected) Shimura variety S = S(Γ) = Γ\X. If HG is a connected semisimple subgroup such that H(\( \mathbb{R} \)) / K is maximal compact, then Y = H(\( \mathbb{R} \))/K is a Hermitian symmetric subdomain of X. For each gG(\( \mathbb{Q} \)) one can construct a connected Shimura variety S(H, g) = (H(\( \mathbb{Q} \)) ∩ g −1Γg)\Y and a natural holomorphic map j g : S(H, g) → S induced by the map H(\( \mathbb{A} \)) → G(\( \mathbb{A} \)), hgh. Let us assume that G is anisotropic, which implies that S and S(H, g) are compact. Then, for each positive integer k, the map j g induces a restriction map
$$ R_{g} :H^{k} {\left( {S,\mathbb{C}} \right)} \to H^{k} {\left( {S{\left( {H,g} \right)},\mathbb{C}} \right)}. $$

In this paper we focus on classical Hermitian domains and give explicit criterions for the injectivity of the product of the maps R g (for g running through G(\( \mathbb{Q} \))) when restricted to the strongly primitive (in the sense of Vogan and Zuckerman) part of the cohomology. In the holomorphic case we recover previous results of Clozel and Venkataramana [CV]. We also derive applications of our results to the proofs of new cases of the Hodge conjecture and of new results on the vanishing of the cohomology of some particular Shimura variety.

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References

  1. [A]
    G. Anderson, Theta functions and holomorphic differential forms on compact quotients of bounded symmetric domains, Duke Math. J. 50 (1983), 1137–1170.MATHCrossRefMathSciNetGoogle Scholar
  2. [B]
    N. Bergeron, Premier nombre de Betti et spectre du laplacien de certaines variétés hyperboliques, Enseign. Math. 46 (2000), 109–137.MATHMathSciNetGoogle Scholar
  3. [BV]
    N. Bergeron, T. N. Venkataramana, A note on the rational structure of the cohomology ring of a Shimura variety, Note non publiée disponible sur http://www.math.jussieu.fr/∼bergeron.
  4. [BHC]
    A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.CrossRefMathSciNetGoogle Scholar
  5. [BW]
    A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Princeton University Press, Princeton, NJ, 1980.MATHGoogle Scholar
  6. [C]
    S. S. Chern, On a generalization of Kaehler geometry, in: Algebraic Geometry and Topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, pp. 103–121.Google Scholar
  7. [Cl]
    L. Clozel, On the cohomology of Kottwitz’s arithmetic varieties, Duke Math. J. 72 (1993), 757–795.MATHCrossRefMathSciNetGoogle Scholar
  8. [COU]
    L. Clozel, H. Oh, E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), 327–351.MATHCrossRefMathSciNetGoogle Scholar
  9. [CV]
    L. Clozel, T. N. Venkataramana, Restriction of the holomorphic cohomology of a Shimura variety to a smaller Shimura variety, Duke Math. J. 95 (1998), 51–106.MATHCrossRefMathSciNetGoogle Scholar
  10. [D]
    P. Deligne, Travaux de Shimura, in: Séminaire Bourbaki, 23ème année (1970/71), Lecture Notes in Mathematics, Vol. 244, Springer-Verlag, New York, 1971, pp. 123–165.Google Scholar
  11. [F]
    W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.MATHGoogle Scholar
  12. [GGPS]
    И. М. Гельфанд, М. И. Граев, И. И. Пятецкий-Шапиро, Теория представлеий и автоморфные функции, Наука, М., 1966. English transl.: I. M. Gelfand, M. I. Graev, I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Academic Press, New York, 1969.Google Scholar
  13. [HL]
    M. Harris, J. S. Li, A Lefschetz property for subvarieties of Shimura varieties, J. Algebraic Geom. 7 (1998), 77–122.MATHMathSciNetGoogle Scholar
  14. [HW]
    R. Hotta, N. R. Wallach, On Matsushima’s formula for the Betti numbers of a locally symmetric space, Osaka J. Math. 12 (1975), 419–431.MATHMathSciNetGoogle Scholar
  15. [K]
    B. Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. 77 (1963), 72–144.CrossRefMathSciNetGoogle Scholar
  16. [Ku]
    S. Kumaresan, The canonical K-types of irreducible (\( \mathfrak{g} \), K)-modules with nonzero cohomology, Invent Math. 59 (1980), 1–11.MATHCrossRefMathSciNetGoogle Scholar
  17. [L]
    J.-S. Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428 (1992), 177–217.MATHMathSciNetGoogle Scholar
  18. [M]
    L. Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénére-scence, Cours Spécialisés, Vol. 3, Société Mathématique de France, Paris, 1998.Google Scholar
  19. [Ma]
    Y. Matsushima, A formula for the Betti numbers of compact locally symmetric Riemannian manifolds, J. Differential Geom. 1 (1967), 99–109.MATHMathSciNetGoogle Scholar
  20. [O]
    T. Oda, A note on the Albanese of an arithmetic quotient of the complex hyperball, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 481–486.MATHMathSciNetGoogle Scholar
  21. [P1]
    R. Parthasarathy, Criteria for the unitarisability of some highest weight modules, Proc. Indian Acad. Sci. 89 (1980), 1–24.MATHCrossRefGoogle Scholar
  22. [P2]
    R. Parthasarathy, Holomorphic forms in Γ\G/K and Chern classes, Topology 21 (1982), 157–178.MATHCrossRefMathSciNetGoogle Scholar
  23. [PR]
    В. Платонов, А. Рапинчук, Алгебраические группы и теория чисел, Наука, М., 1991. English transl.: V. Platonov, A. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, Vol. 139, Academic Press, Boston, MA, 1994.Google Scholar
  24. [R]
    J. Rohlfs, Projective limits of locally symmetric spaces and cohomology, J. Reine Angew. Math. 479 (1996), 149–182.MATHMathSciNetGoogle Scholar
  25. [S]
    I. Satake, Holomorphic imbeddings of symmetric domains into Siegel space, Amer. J. Math. 87 (1965), 425–461.MATHCrossRefMathSciNetGoogle Scholar
  26. [V]
    T. N. Venkataramana, Abelianness of Mumford-Tate groups associated to some unitary groups, Compositio Math. 122 (2000), 223–242.MATHCrossRefMathSciNetGoogle Scholar
  27. [V2]
    T. N. Venkataramana, Cohomology of compact locally symmetric spaces, Compositio Math. 125 (2001), 221–253.MATHCrossRefMathSciNetGoogle Scholar
  28. [V3]
    T. N. Venkataramana, Some remarks on cycle classes on Shimura varieties, J. Ramanujan Math. Soc. 16 (2001), 309–322.MATHMathSciNetGoogle Scholar
  29. [V4]
    T. N. Venkataramana, On cycles on compact locally symmetric varieties, Monatsh. Math. 135 (2002), 221–244.MATHCrossRefMathSciNetGoogle Scholar
  30. [VZ]
    D. Vogan, G. Zuckerman, Unitary representations with cohomology, Compositio Math. 53 (1984), 51–90.MATHMathSciNetGoogle Scholar
  31. [Z]
    A. V. Zelevinsky, A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence, J. Algebra 69 (1981), 82–94.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu U. M. R. 7586 du CNRSUniversité Pierre et Marie CurieParis Cedex 05France

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