Skip to main content Accessibility help
×
Home

Residues of Eisenstein series and the automorphic cohomology of reductive groups

  • Harald Grobner (a1)

Abstract

Let $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $ . In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$ , $q\lt {q}_{ \mathsf{res} } $ , for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $ , the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $ , ${C}_{n} $ , ${D}_{n} $ .

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Residues of Eisenstein series and the automorphic cohomology of reductive groups
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Residues of Eisenstein series and the automorphic cohomology of reductive groups
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Residues of Eisenstein series and the automorphic cohomology of reductive groups
      Available formats
      ×

Copyright

References

Hide All
[BR10]Badulescu, I. A. and Renard, D., Unitary Dual of GL$(n)$ at Archimedean places and global Jacquet–Langlands correspondence, Comput. Math. 145 (2010), 11151164.
[Bor07]Borel, A., Automorphic forms on reductive groups, in Automorphic forms and applications, IAS/Park City Mathematics Series, vol. 12, Utah, 2002, eds Sarnak, P. and Shahidi, F. (American Mathematical Society, Providence, RI, 2007), 540.
[Bor74]Borel, A., Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Supér. 7 (1974), 235272.
[BC83]Borel, A. and Casselman, W., ${L}^{2} $-Cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), 625647.
[BJ79]Borel, A. and Jacquet, H., Automorphic forms and automorphic representations, Proceedings of Symposia in Pure Mathematics, vol. XXXIII, part I (American Mathematical Society, Providence, RI, 1979), 189202.
[BLS96]Borel, A., Labesse, J.-P. and Schwermer, J., On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields, Comput. Math. 102 (1996), 140.
[BW80]Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Mathematics Studies, vol. 94 (Princeton University Press, Princeton, NJ, 1980).
[Clo90]Clozel, L., Motifs et Formes Automorphes: Applications du Principe de Fonctorialité, in Automorphic forms, Shimura varieties, and L-functions, vol. I, Ann Arbor, MI, 1988, Perspectives on Mathematics, vol. 10, eds Clozel, L. and Milne, J. S. (Academic Press, Boston, MA, 1990), 77159.
[Clo93]Clozel, L., On the cohomology of Kottwitz’s arithmetic varieties, Duke Math. J. 72 (1993), 757795.
[Enr79]Enright, T. J., Relative Lie algebra cohomology and unitary representations of complex Lie groups, Duke Math. J. 46 (1979), 513525.
[Fra98]Franke, J., Harmonic analysis in weighted ${L}_{2} $-spaces, Ann. Sci. Éc. Norm. Supér. 2 (1998), 181279.
[Fra08]Franke, J., A topological model for some summand of the Eisenstein cohomology of congruence subgroups, in Eisenstein series and applications, Progress in Mathematics, vol. 258, eds Gan, W. T., Kudla, S. S. and Tschinkel, Y. (Birkhäuser, Boston, 2008), 2785.
[FS98]Franke, J. and Schwermer, J., A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 311 (1998), 765790.
[GG12]Gotsbacher, G. and Grobner, H., On the Eisenstein cohomology of odd orthogonal groups, Forum Math. (2012), to appear, doi:10.1515/FORM.2011.118.
[GG13]Grbac, N. and Grobner, H., The residual Eisenstein cohomology of $S{p}_{4} $ over a totally real number field, Trans. Amer. Math. Soc. (2013), to appear.
[GS11]Grbac, N. and Schwermer, J., On residual cohomology classes attached to relative rank one Eisenstein series for the symplectic group, Int. Math. Res. Not. IMRN 7 (2011), 16541705.
[Gro13]Grobner, H., Automorphic Forms, Cohomology and CAP Representations. The Case ${\mathrm{GL} }_{2} $ over a definite quaternion algebra, J. Ramanujan Math. Soc. 28 (2013), 1943.
[Har87]Harder, G., Eisenstein cohomology of arithmetic groups. The case $G{L}_{2} $, Invent. Math. 89 (1987), 37118.
[Har75a]Harder, G., On the cohomology of $SL(2, \mathfrak{O})$, in Lie groups and their representations: Proc. of the summer school on group representations, Budapest, 1971, ed. Gelfand, I. M. (Halsted, New York, 1975), 139150.
[Har75b]Harder, G., On the cohomology of discrete arithmetically defined groups, in Discrete subgroups of Lie groups and applications to moduli, Papers presented at the Bombay Colloquium, Bombay, 1973 (Oxford University Press, Oxford, 1975), 129160.
[Kum80]Kumaresan, S., On the canonical $k$-types in the irreducible unitary $g$-modules with non-zero relative cohomology, Invent. Math. 59 (1980), 111.
[Lan76]Langlands, R. P., On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, vol. 544 (Springer, Berlin, 1976).
[LS04]Li, J.-S. and Schwermer, J., On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 (2004), 141169.
[MW89]Mœglin, C. and Waldspurger, J.-L., Le spectre résiduel de $GL(n)$, Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), 605674.
[MW95]Mœglin, C. and Waldspurger, J.-L., Spectral decomposition and Eisenstein series (Cambridge University Press, Cambridge, 1995).
[RS11]Rohlfs, J. and Speh, B., Pseudo Eisenstein forms and the cohomology of arithmetic groups III: residual cohomology classes, in On certain L-functions: Conference in Honor of Freydoon Shahidi, Purdue University, West Lafayette, Indiana, July 23–27, 2007, eds Arthur, J., Codgell, J. W., Gelbart, S., Goldberg, D., Ramakrishnan, D. and Yu, J.-K. (American Mathematical Society, Providence, RI, 2011), 501524.
[Sch83]Schwermer, J., Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Mathematics, vol. 988 (Springer, 1983).
[Sch94]Schwermer, J., Eisenstein series and cohomology of arithmetic groups: the generic case, Invent. Math. 116 (1994), 481511.
[VZ84]Vogan, D. A. Jr. and Zuckerman, G. J., Unitary representations with nonzero cohomology, Comput. Math. 53 (1984), 5190.
[Wal84]Wallach, N., On the constant term of a square integrable automorphic form, in Operator algebras and group representations, vol. II, Neptun, 1980, Monographs and Studies in Mathematics, vol. 18 (Pitman, Boston, MA, London, 1984), 227237.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Residues of Eisenstein series and the automorphic cohomology of reductive groups

  • Harald Grobner (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed