Duke Mathematical Journal

On the Eisenstein cohomology of arithmetic groups

Jian-Shu Li and Joachim Schwermer

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The cohomology $H ^* (\Gamma, E)$ of an arithmetic subgroup $\Gamma$ of a connected reductive algebraic group $G$ defined over some algebraic number field $F$ can be interpreted in terms of the automorphic spectrum of $\Gamma$. With this framework in place, there is a sum decomposition of the cohomology into the cuspidal cohomology (i.e., classes represented by cuspidal automorphic forms for $G$) and the so-called Eisenstein cohomology constructed as the span of appropriate residues or derivatives of Eisenstein series attached to cuspidal automorphic forms on the Levi components of proper parabolic $F$-subgroups of $G$. The main objective of this paper is to isolate a specific structural part in the Eisenstein cohomology. This pertains to regular Eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered. It is shown that the cohomological degree of these classes is bounded from below by the constant $q_0(G(\mathbb{R}))=((1/2) [\dim X_{G(\mathbb{R})}-(\rk (G(\mathbb{R}))-\rk(K_{\mathbb{R}}))]$, where $K_{\mathbb{R}}$ denotes a maximal compact subgroup of the real Lie group $G(\mathbb{R})$, where $X_{G(\mathbb{R})}$ is the associated symmetric space. This investigation has various applications. One of these is a vanishing result for the cohomology in the generic case (i.e., where the representation determining the coefficient system $E$ has regular highest weight) in degrees below $q_0(G(\mathbb{R}))$. This is a sharp bound depending only on the underlying real Lie group $G(\mathbb{R})$ (Corollary 5.6, Proposition 5.8). This result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular Eisenstein cohomology classes.

Article information

Duke Math. J. Volume 123, Number 1 (2004), 141-169.

First available in Project Euclid: 13 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F75: Cohomology of arithmetic groups
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]


Li, Jian-Shu; Schwermer, Joachim. On the Eisenstein cohomology of arithmetic groups. Duke Math. J. 123 (2004), no. 1, 141--169. doi:10.1215/S0012-7094-04-12315-2. http://projecteuclid.org/euclid.dmj/1084479321.

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  • A. Borel, Regularization theorems in Lie algebra cohomology: Applications, Duke Math. J. 50 (1983), 605--623.
  • A. Borel, J.-P. Labesse, and J. Schwermer, On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields, Compositio Math. 102 (1996), 1--40.
  • A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, 2000.
  • J. Franke, Harmonic analysis in weighted $L_2$-spaces, Ann. Sci. École Norm. Sup. (4) 31 (1998), 181--279.
  • J. Franke and J. Schwermer, A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 331 (1998), 765--790.
  • G. Harder, ``On the cohomology of discrete arithmetically defined groups'' in Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, Bombay, 1975, 129--160.
  • --. --. --. --., ``On the cohomology of $\SL (2, \mathcalO)$'' in Lie Groups and Their Representations, (Budapest, 1971), Halsted, New York, 1975, 139--150.
  • --. --. --. --., Eisenstein cohomology of arithmetic groups: The case $\GL_2$, Invent. Math. 89 (1987), 37--118.
  • --. --. --. --., ``Some results on the Eisenstein cohomology of arithmetic subgroups of $\GL_n$'' in Cohomology of Arithmetic Groups and Automorphic Forms (Luminy-Marseille, 1989), Lecture Notes in Math. 1447, Springer, Berlin, 1990, 85--153.
  • Harish-Chandra, Automorphic Forms on Semisimple Lie Groups, Lecture Notes in Math. 62, Springer, Berlin, 1968.
  • B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329--387.
  • J.-P. Labesse and J. Schwermer, On liftings and cusp cohomology of arithmetic groups, Invent. Math. 83 (1986), 383--401.
  • R. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 554, Springer, Berlin, 1976.
  • --. --. --. --., ``On the classification of irreducible representations of real algebraic groups'' in Representation Theory and Harmonic Analysis on Semisimple Liegroups, Math. Surveys Monogr. 31, Amer. Math. Soc., Providence, 1989, 101--170.
  • --------, letter to Borel, October 25, 1972.
  • J.-S. Li and J. Schwermer, Constructions of automorphic forms and related cohomology classes for arithmetic subgroups of $G_2$, Compositio Math. 87 (1993), 45--78.
  • --. --. --. --., ``Automorphic representations and cohomology of arithmetic groups'' in Challenges for the 21st Century (Singapore, 2000), World Sci., River Edge, N.J., 2001, 102--137.
  • C. Moeglin and J.-L. Waldspurger, Décomposition spectrale et séries d'Eisenstein: une paraphrase de l'écriture, Progr. Math. 113, Birkhäuser, Basel, 1993.
  • I. T. Piatetski-Shapiro, ``Euler subgroups'' in Lie Groups and Their Representations (Budapest, 1971), Halsted, New York, 1975, 597--620.
  • J. Rohlfs, Projective limits of locally symmetric spaces and cohomology, J. Reine Angew. Math. 479 (1996), 149--182.
  • L. Saper, ``On the cohomology of locally symmetric spaces and of their compactifications'' in Current Developments in Mathematics 2002, International Press, Somerville, Mass., 2004.
  • --------, $L$-modules and the conjecture of Rapoport and Goresky-MacPherson, preprint.
  • --------, $L$-modules and Micro-support, preprint.
  • J. Schwermer, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Math. 988, Springer, Berlin, 1983.
  • --. --. --. --., Holomorphy of Eisenstein series at special points and cohomology of arithmetic subgroups of $\SL_n(\mathbbQ)$, J. Reine Angew. Math. 364 (1986), 193--220.
  • --. --. --. --., Eisenstein series and cohomology of arithmetic groups: The generic case, Invent. Math. 116 (1994), 481--511.
  • --. --. --. --., On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties, Forum Math. 7 (1995), 1--28.
  • J. A. Shalika, The multiplicity one theorem for $\GL_n$, Ann. of Math. (2) 100 (1974), 171--193.
  • B. Speh, Unitary representations of $\GL(n,\mathbbR)$ with nontrivial $(\mathfrakg, K)$-cohomology, Invent. Math. 71 (1983), 443--465.
  • J. Tilouine and E. Urban, Several-variable $p$-adic families of Siegel Hilbert cusp eigensystems and their Galois representations, Ann. Sci. École Norm. Sup. (4) 32 (1999), 499--574.
  • D. A. Vogan and G. J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), 51--90.
  • N. R. Wallach, ``On the constant term of a square integrable automorphic form'' in Operator Algebras and Group Representations, II (Neptun, Romania, 1980), Monogr. Stud. Math. 18, Pitman, Boston, 1984, 227--237.