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Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Published online by Cambridge University Press:  23 November 2009

Harald Grobner*
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria (email: harald.grobner@univie.ac.at)
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Abstract

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Let G be the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8 of rank four. The cohomology of the space of automorphic forms on G has a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomology HqEis(G,E) of G in the case of regular coefficients E. It is spanned only by holomorphic Eisenstein series. For non-regular coefficients E we really have to detect the poles of our Eisenstein series. Since G is not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297–355; F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolic P0 of G. Having collected this information, we determine the square-integrable Eisenstein cohomology supported by P0 with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Borel, A., Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235272.CrossRefGoogle Scholar
[2]Borel, A. and Casselman, W., L 2-cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), 625647.CrossRefGoogle Scholar
[3]Borel, A., Labesse, J.-P. and Schwermer, J., On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields, Compositio Math. 102 (1996), 140.Google Scholar
[4]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).Google Scholar
[5]Flath, D., Decomposition of representations into tensor products, Proceedings of Symposia in Pure Mathematics, vol. XXXIII, part I (American Mathematical Society, Providence, RI, 1979), 179183.Google Scholar
[6]Franke, J., Harmonic analysis in weighted L 2-spaces, Ann. Sci. École Norm. Sup. (4) 31 (1998), 181279.CrossRefGoogle Scholar
[7]Franke, J. and Schwermer, J., A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 311 (1998), 765790.CrossRefGoogle Scholar
[8]Gelbart, S. and Jacquet, H., Forms on GL(2) from the analytic point of view, Proceedings of Symposia in Pure Mathematics, vol. XXXIII, part I (American Mathematical Society, Providence, RI, 1979), 213251.Google Scholar
[9]Gelfand, I. M., Graev, M. I. and Piatetski-Shapiro, I., Representation theory and automorphic functions (W. B. Saunders Company, Philadelphia, PA, 1969).Google Scholar
[10]Grbac, N., Correspondence between the residual spectrum of rank two split classical groups and their inner forms, in Functional analysis IX - proceedings of the postgraduate school and conference held at the Inter-University centre, Dubrovnik, Croatia, 15–23 June, 2005, Various Publication Series, vol. 58, eds Muic, G. and Hoffman-Jørgensen, J. (University of Aarhus, Aarhus, 2007), 4457.Google Scholar
[11]Grbac, N., On a relation between residual spectra of split classical groups and their inner forms, Canad. J. Math., Preprint (2009), to appear.CrossRefGoogle Scholar
[12]Grobner, H., The automorphic cohomology and the residual spectrum of hermitian groups of rank one, Int. J. Math. (2009), to appear, electronically available athttp://homepage.univie.ac.at/harald.grobner/publication.html.CrossRefGoogle Scholar
[13]Harder, G., On the cohomology of SL(2,𝔒), in Lie groups and their representations, Proc. of the summer school on group representations, ed. I. M. Gelfand (Adam Hilger Ltd., London, 1975), 139–150.Google Scholar
[14]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, 1973 (Oxford University Press, Oxford, 1975), 129160.Google Scholar
[15]Jacquet, H., Automorphic forms on GL(2), Part II, Lecture Notes in Mathematics, vol. 278 (Springer, Berlin, 1972).CrossRefGoogle Scholar
[16]Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[17]Kim, H. H., The residual spectrum of Sp4, Compositio Math. 99 (1995), 129151.Google Scholar
[18]Kostant, B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329387.CrossRefGoogle Scholar
[19]Langlands, R. P., Euler products (Yale University Press, New Haven, CT, 1971).Google Scholar
[20]Langlands, R. P., Letter to A. Borel, dated October 25, (1972).Google Scholar
[21]Langlands, R. P., On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, vol. 544 (Springer, Berlin, 1976).CrossRefGoogle Scholar
[22]Langlands, R. P., On the notion of an automorphic representation. A supplement to the preceding paper, Proceedings of Symposia in Pure Mathematics, vol. XXXIII, part I (American Mathematical Society, Providence, RI, 1979), 189202.Google Scholar
[23]Li, J.-S. and Schwermer, J., On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 (2004), 142169.CrossRefGoogle Scholar
[24]Margulis, G. A., Discrete subgroups of semisimple Lie groups (Springer, Berlin, 1991).CrossRefGoogle Scholar
[25]Mœglin, C. and Waldspurger, J.-L., Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4) 22 (1989), 605674.CrossRefGoogle Scholar
[26]Mœglin, C. and Waldspurger, J.-L., Spectral decomposition and Eisenstein series (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[27]Muić, G. and Savin, G., Complementary series for Hermitian quaternionic groups, Canad. Math. Bull. 43 (2000), 9099.CrossRefGoogle Scholar
[28]Platonov, V. and Rapinchuk, A., Algebraic groups and number theory (London Academic Press, London, 1993).Google Scholar
[29]Raghunathan, M. S., Cohomology of arithmetic subgroups of algebraic groups: I, Ann. of Math. (2) 86 (1967), 409424.CrossRefGoogle Scholar
[30]Rohlfs, J., Projective limits of locally symmetric spaces and cohomology, J. Reine Angew. Math. 479 (1996), 149182.CrossRefGoogle Scholar
[31]Rohlfs, J. and Speh, B., Representations with cohomology in the discrete spectrum of subgroups of SO(n,1)(ℤ) and Lefschetz numbers, Ann. Sci. École Norm. Sup. (4) 20 (1987), 89136.CrossRefGoogle Scholar
[32]Schwermer, J., Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Mathematics, vol. 988 (Springer, Berlin, 1983).CrossRefGoogle Scholar
[33]Schwermer, J., Eisenstein series and cohomology of arithmetic groups: the generic case, Inventiones Mathematicae, vol. 116 (Springer, Berlin, 1994), 481511.Google Scholar
[34]Shahidi, F., On Certain L-functions, Amer. J. Math. 103 (1981), 297355.CrossRefGoogle Scholar
[35]Shahidi, F., On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547584.CrossRefGoogle Scholar
[36]Tate, J., Fourier analysis in number fields and Hecke’s zeta-functions, in Algebraic number theory eds Cassels, J. W. S. and Fröhlich, A. (Academic Press, Boston, MA, 1967), 305347.Google Scholar
[37]Vogan, D. A. Jr. and Zuckerman, G. J., Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), 5190.Google Scholar
[38]Yasuda, T., The residual spectrum of inner forms of Sp(2), Pacific J. Math. 232 (2007), 471490.CrossRefGoogle Scholar