## On the Eisenstein cohomology of arithmetic groups.(English)Zbl 1057.11031

Let $$G$$ be a connected semisimple algebraic group over $${\mathbb Q}$$ and $$\Gamma\subseteq G({\mathbb R})$$ a torsion-free arithmetic subgroup. Denote by $$K$$ a maximal compact subgroup of $$G({\mathbb R})$$ and by $$X$$ the quotient space $$G({\mathbb R}) / K.$$ If $$(\nu, E)$$ is a finite-dimensional representation of $$G({\mathbb R}),$$ then the cohomology $$H^\ast (\Gamma, E)$$ is that of a local system $${\mathcal E}$$ defined on $$\Gamma\backslash X.$$ This in turn can be described by certain automorphic data, and one gets the notion of Eisenstein cohomology. This is a part of $$H^\ast(\Gamma\backslash X, {\mathcal E})$$ corresponding to cuspidal automorphic forms on Levi components of (proper) parabolic subgroups of $$G$$. This decomposition is achieved in [J. Franke and J. Schwermer, Math. Ann. 311, No. 4, 765–790 (1998; Zbl 0924.11042)] as a result of Franke’s proof [Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, No. 2, 181–279 (1998; Zbl 0938.11026)] of a conjecture of Borel.
The authors now aim at refining this decomposition. In particular, they show that if an Eisenstein-class is regular and comes from a cuspidal automorphic representation whose archimedian component is tempered then its cohomological degree is bounded from below by $(1/2) [\dim X - \text{rk} (G({\mathbb R})) + \text{rk} (K)].$ This result gives rise to new vanishing theorems. Moreover, if the $${\mathbb Q}$$-rank of $$G$$ is positive and $$(\nu,E)$$ has regular highest weight, then in degrees larger than $$(1/2) [\dim X + \text{ rk} (G({\mathbb R})) - \text{ rk} (K)]$$ the cohomology of $$\mathcal E$$ is spanned by regular Eisenstein classes.
The results are achieved in a more general setting and the adelic language is being used throughout.

### MSC:

 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F75 Cohomology of arithmetic groups 22E40 Discrete subgroups of Lie groups

### Keywords:

Eisenstein cohomology; harmonic analysis

### Citations:

Zbl 0924.11042; Zbl 0938.11026
Full Text:

### References:

 [1] A. Borel, Regularization theorems in Lie algebra cohomology: Applications , Duke Math. J. 50 (1983), 605–623. · Zbl 0528.22010 [2] 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. · Zbl 0853.11044 [3] 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. · Zbl 0980.22015 [4] J. Franke, Harmonic analysis in weighted $$L_2$$-spaces , Ann. Sci. École Norm. Sup. (4) 31 (1998), 181–279. · Zbl 0938.11026 [5] 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. · Zbl 0924.11042 [6] 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. · Zbl 0317.57022 [7] –. –. –. –., “On the cohomology of $$\SL (2, \mathcalO)$$” in Lie Groups and Their Representations, (Budapest, 1971) , Halsted, New York, 1975, 139–150. [8] –. –. –. –., Eisenstein cohomology of arithmetic groups: The case $$\GL_2$$ , Invent. Math. 89 (1987), 37–118. · Zbl 0629.10023 [9] –. –. –. –., “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. · Zbl 0719.11034 [10] Harish-Chandra, Automorphic Forms on Semisimple Lie Groups , Lecture Notes in Math. 62 , Springer, Berlin, 1968. · Zbl 0186.04702 [11] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem , Ann. of Math. (2) 74 (1961), 329–387. JSTOR: · Zbl 0134.03501 [12] J.-P. Labesse and J. Schwermer, On liftings and cusp cohomology of arithmetic groups , Invent. Math. 83 (1986), 383–401. · Zbl 0581.10013 [13] R. Langlands, On the Functional Equations Satisfied by Eisenstein Series , Lecture Notes in Math. 554 , Springer, Berlin, 1976. · Zbl 0332.10018 [14] –. –. –. –., “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. · Zbl 0741.22009 [15] ——–, letter to Borel, October 25, 1972. [16] 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. · Zbl 0795.11025 [17] –. –. –. –., “Automorphic representations and cohomology of arithmetic groups” in Challenges for the 21st Century (Singapore, 2000) , World Sci., River Edge, N.J., 2001, 102–137. · Zbl 1081.11507 [18] C. Moeglin and J.-L. Waldspurger, Décomposition spectrale et séries d’Eisenstein: une paraphrase de l’écriture , Progr. Math. 113 , Birkhäuser, Basel, 1993. [19] I. T. Piatetski-Shapiro, “Euler subgroups” in Lie Groups and Their Representations (Budapest, 1971) , Halsted, New York, 1975, 597–620. [20] J. Rohlfs, Projective limits of locally symmetric spaces and cohomology , J. Reine Angew. Math. 479 (1996), 149–182. · Zbl 0876.22021 [21] L. Saper, “On the cohomology of locally symmetric spaces and of their compactifications” in Current Developments in Mathematics 2002 , International Press, Somerville, Mass., 2004. [22] ——–, $$L$$-modules and the conjecture of Rapoport and Goresky-MacPherson , · Zbl 1083.11033 [23] ——–, $$L$$-modules and Micro-support , [24] J. Schwermer, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen , Lecture Notes in Math. 988 , Springer, Berlin, 1983. · Zbl 0506.22015 [25] –. –. –. –., Holomorphy of Eisenstein series at special points and cohomology of arithmetic subgroups of $$\SL_n(\mathbbQ)$$ , J. Reine Angew. Math. 364 (1986), 193–220. · Zbl 0571.10032 [26] –. –. –. –., Eisenstein series and cohomology of arithmetic groups: The generic case , Invent. Math. 116 (1994), 481–511. · Zbl 0807.11031 [27] –. –. –. –., On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties , Forum Math. 7 (1995), 1–28. · Zbl 0829.11031 [28] J. A. Shalika, The multiplicity one theorem for $$\GL_n$$ , Ann. of Math. (2) 100 (1974), 171–193. JSTOR: · Zbl 0316.12010 [29] B. Speh, Unitary representations of $$\GL(n,\mathbbR)$$ with nontrivial $$(\mathfrakg, K)$$-cohomology , Invent. Math. 71 (1983), 443–465. · Zbl 0505.22015 [30] 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. · Zbl 0991.11016 [31] D. A. Vogan and G. J. Zuckerman, Unitary representations with nonzero cohomology , Compositio Math. 53 (1984), 51–90. · Zbl 0692.22008 [32] 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. · Zbl 0554.22004
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