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Boundary and Eisenstein cohomology of \(\mathrm{SL}_3(\mathbb{Z})\). (English) Zbl 1473.11111

Summary: In this article, several cohomology spaces associated to the arithmetic groups \(\text{SL}_3({\mathbb{Z}})\) and \(\text{GL}_3({\mathbb{Z}})\) with coefficients in any highest weight representation \({\mathcal{M}}_\lambda\) have been computed, where \(\lambda\) denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in \({\mathcal{M}}_\lambda \). When \({\mathcal{M}}_\lambda\) is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in \({\mathcal{M}}_\lambda \). In particular, for such a large class of representations we can explicitly describe the cohomology of these two arithmetic groups. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification and their Euler characteristic with coefficients in \({\mathcal{M}}_\lambda \). At the end, we employ our study to discuss the existence of ghost classes.

MSC:

11F75 Cohomology of arithmetic groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F22 Relationship to Lie algebras and finite simple groups
11F06 Structure of modular groups and generalizations; arithmetic groups
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[1] Bajpai, J., Moya Giusti, M.: Ghost classes in \({\mathbb{Q}} \)-rank two orthogonal Shimura varieties. Math. Z. (2020) · Zbl 1451.14080
[2] Bass, H.; Milnor, J.; Serre, J-P, Solution of the congruence subgroup problem for \({{\rm SL}}_n\,(n\ge 3)\) and \({{\rm Sp}}_{2n}\,(n\ge 2)\), Inst. Hautes Études Sci. Publ. Math., 33, 59-137 (1967) · Zbl 0174.05203 · doi:10.1007/BF02684586
[3] Borel, A.: Cohomology and spectrum of an arithmetic group. In: Operator algebras and group representations, vol. I (Neptun, 1980). Monographs Studies in Mathematics, vol. 17, pp. 28-45. Pitman, Boston (1984) · Zbl 0525.22013
[4] Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436-491 (1973) (avec un appendice: arrondissement des variétés à coins, par A. Douady et L. Hérault) · Zbl 0274.22011
[5] Brown, K.S.: Cohomology of groups. In: Graduate Texts in Mathematics, vol. 87. Springer, New York (1994) (corrected reprint of the 1982 original)
[6] Chiswell, IM, Euler characteristics of groups, Math. Z., 147, 1, 1-11 (1976) · Zbl 0304.20022 · doi:10.1007/BF01214269
[7] Franke, J., Harmonic analysis in weighted \(L_2\)-spaces, Ann. Sci. École Norm. Sup. (4), 31, 2, 181-279 (1998) · Zbl 0938.11026 · doi:10.1016/S0012-9593(98)80015-3
[8] Fulton, W., Harris, J.: Representation theory. In: Graduate Texts in Mathematics, vol. 129. Springer, New York (1991) (a first course, readings in mathematics) · Zbl 0744.22001
[9] Gelfand, I.M., Cetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk. SSSR (N.S.) 71, 825-828 (1950) · Zbl 0037.15301
[10] Harder, G., A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. École Norm. Sup., 4, 4, 409-455 (1971) · Zbl 0232.20088 · doi:10.24033/asens.1217
[11] Harder, G., Eisenstein cohomology of arithmetic groups. The case \({{\rm GL}}_2\), Invent. Math., 89, 1, 37-118 (1987) · Zbl 0629.10023 · doi:10.1007/BF01404673
[12] Harder, G.: Some results on the Eisenstein cohomology of arithmetic subgroups of \({\rm GL}_n\). In: Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989). Lecture Notes in Mathematics, vol. 1447, pp. 85-153. Springer, Berlin (1990) · Zbl 0719.11034
[13] Harder, G.: Arithmetic aspects of rank one Eisenstein cohomology. In: Cycles, motives and Shimura varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 21, pp. 131-190. Tata Institute of Fundamental Research, Mumbai (2010) · Zbl 1294.11078
[14] Harder, G.: The Eisenstein motive for the cohomology of \({{\rm GSp}}_2(\mathbb{Z})\). In: Geometry and arithmetic, EMS Series of Congress Reports, pp. 143-164. European Mathematical Society, Zürich (2012) · Zbl 1317.11056
[15] Harder, G.: The cohomology of arithmetic Groups. http://www.math.uni-bonn.de/people/harder/Manuscripts/buch/Volume-III.Feb-26-2020.pdf (in preparation) (2020) · Zbl 0629.10023
[16] Harder, G., Raghuram, A.: Eisenstein cohomology for \({\rm GL}_N\) and the special values of Rankin-Selberg \(L\)-functions. In: Annals of Mathematics Studies, vol. 203. Princeton University Press, Princeton (2020) · Zbl 1466.11001
[17] Horozov, I.: Euler characteristics of arithmetic groups. Thesis (Ph.D.), Brown University.ProQuest LLC, Ann Arbor (2004) · Zbl 1114.20025
[18] Horozov, I., Euler characteristics of arithmetic groups, Math. Res. Lett., 12, 2-3, 275-291 (2005) · Zbl 1114.20025 · doi:10.4310/MRL.2005.v12.n3.a1
[19] Horozov, I., Cohomology of \(GL_4(\mathbb{Z})\) with nontrivial coefficients, Math. Res. Lett., 21, 5, 1111-1136 (2014) · Zbl 1355.11064 · doi:10.4310/MRL.2014.v21.n5.a9
[20] Kewenig, A., Rieband, T.: Geisterklassen im bild der borelabbildung fur symplektische und orthogonale gruppen. Master’s Thesis (1997)
[21] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. Math., 2, 74, 329-387 (1961) · Zbl 0134.03501 · doi:10.2307/1970237
[22] Lee, R.; Schwermer, J., Cohomology of arithmetic subgroups of \({\rm SL}_3\) at infinity, J. Reine Angew. Math., 330, 100-131 (1982) · Zbl 0463.57014
[23] Moya Giusti, M., Ghost classes in the cohomology of the Shimura variety associated to \(GSp_4\), Proc. Am. Math. Soc., 146, 6, 2315-2325 (2018) · Zbl 1423.14178 · doi:10.1090/proc/13788
[24] Moya Giusti, M., On the existence of ghost classes in the cohomology of the Shimura variety associated to \(GU(2,2)\), Math. Res. Lett., 25, 4, 1227-1249 (2018) · Zbl 1444.14053 · doi:10.4310/MRL.2018.v25.n4.a9
[25] Rohlfs, J., Projective limits of locally symmetric spaces and cohomology, J. Reine Angew. Math., 479, 149-182 (1996) · Zbl 0876.22021 · doi:10.1515/crll.1996.479.149
[26] Schwermer, J.: Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen. In: Lecture Notes in Mathematics, vol. 988. Springer, Berlin (1983) · Zbl 0506.22015
[27] Serre, J-P, Cohomologie des groupes discrets, Ann. Math. Stud., 70, 77-169 (1971) · Zbl 0229.57016
[28] Soulé, C., The cohomology of \({\rm SL}_3({ Z})\), Topology, 17, 1, 1-22 (1978) · Zbl 0382.57026 · doi:10.1016/0040-9383(78)90009-5
[29] Wall, CTC, Rational Euler characteristics, Proc. Camb. Philos. Soc., 57, 182-184 (1961) · Zbl 0100.25701 · doi:10.1017/S030500410003499X
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