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Boundary behaviour of special cohomology classes arising from the Weil representation

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  03 July 2012

Jens Funke  and
John Millson
Jens Funke
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Rd, Durham DH1 3LE, UK ([email protected])
John Millson
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA ([email protected])
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Abstract

In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.

In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.

As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

Keywords

Weil representationcohomology of locally symmetric spaces

MSC classification

Secondary: 11F27: Theta series; Weil representation; theta correspondences 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 3 , July 2013 , pp. 571 - 634
Copyright
©Cambridge University Press 2012 

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References

Abramenko, P. and Brown, K., Buildings, theory and applications, GTM, Volume 248 (Springer, 2008).Google Scholar
Bergeron, N., Millson, J. and Moeglin, C., Hodge type theorems for arithmetic manifolds associated to orthogonal groups, preprint (arXiv:math/1110.3049).Google Scholar
Borel, A. and Casselman, W., ${L}^{2} $ cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), 625–647.CrossRefGoogle Scholar
Borel, A. and Ji, L., Compactifications of symmetric and locally symmetric spaces (Birkhäuser, 2006).Google Scholar
Borel, A. and Serre, J.-P., Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491.Google Scholar
Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, Mathematical Surveys and Monographs, Volume 67 (AMS, 2000).Google Scholar
Bourbaki, N., Groupes et Algèbres de Lie, Chapters 4–6. (Hermann, 1981).Google Scholar
Cogdell, J., Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus, J. Reine Angew. Math. 357 (1985), 115–137.Google Scholar
Fulton, W. and Harris, J., Representation theory, a first course, Graduate Texts in Mathematics, Volume 129 (Springer, 1991).Google Scholar
Funke, J., Special cohomology classes for the Weil representation, Proc. of the Conference on Automorphic Forms and Automorphic L-Functions (RIMS Kokyuroku 1617, Kyoto, 2008).Google Scholar
Funke, J. and Millson, J., Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, Amer. J. Math. 128 (2006), 899–948.Google Scholar
Funke, J. and Millson, J., The geometric theta correspondence for Hilbert modular surfaces, preprint (arXiv:math/1108.5305).Google Scholar
Funke, J. and Millson, J., Spectacle cycles with coefficients and modular forms of half-integral weight, in Arithmetic geometry and automorphic forms, Volume in honor of the 60th birthday of Stephen S. Kudla. Advanced Lectures in Mathematics Series (International Press and the Higher Education Press of China, 2011).Google Scholar
Garrett, P., Buildings and classical groups. (Chapman & Hall, 1997).Google Scholar
Goodman, R. and Wallach, N. R., Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, Volume 68 (Cambridge University Press, 1998).Google Scholar
Goresky, M., Harder, G. and MacPherson, R., Weighted cohomology, Invent. Math. 116 (1994), 139–213.Google Scholar
Harder, G., On the cohomology of discrete arithmetically defined groups, Proc. of the Int. Colloq. on Discrete Subgroups of Lie Groups and Appl. to Moduli (Bombay, 1973), pp. 129–160 (Oxford, 1975).Google Scholar
Hirzebruch, F. and Zagier, D., Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113.Google Scholar
Kostant, B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. 74 (1961), 329–387.CrossRefGoogle Scholar
Kudla, S. and Millson, J., The theta correspondence and harmonic forms I, Math. Ann. 274 (1986), 353–378.Google Scholar
Kudla, S. and Millson, J., The theta correspondence and harmonic forms II, Math. Ann. 277 (1987), 267–314.Google Scholar
Kudla, S. and Millson, J., Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Publ. Math. Inst. Hautes Études Sci. 71 (1990), 121–172.Google Scholar
Lebedev, N. N., Special functions and their applications. (Dover, 1972).Google Scholar
Lee, J. M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Volume 218 (Springer, 2002).Google Scholar
Li, J.-S., Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428 (1992), 177–217.Google Scholar
Millson, J., On the first Betti number of a constant negatively curved manifold, Ann. of Math. 104 (1976), 235–247.Google Scholar
Millson, J., The cohomology with local coefficients of compact hyperbolic manifolds, Proceedings of the International Conference on Algebraic Groups and Arithmetic Subgroups, Mumbai (2001) (Narosa Publishing House, international distribution by the AMS).Google Scholar
Millson, J., The cohomology with local coefficients of compact hyperbolic manifolds — long version (arXiv:math/0306296).Google Scholar
Schwermer, J., Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Math., Volume 988 (Springer, 1983).Google Scholar
Shintani, T., On the construction of holomorphic cusp forms of half integral weight, Nagoya math. J. 58 (1975), 83–126.Google Scholar
Speh, B. and Venkataramana, T. N., Construction of some generalised modular symbols, Pure Appl. Math. Q. 1 (2005), 737–754.CrossRefGoogle Scholar
Vogan, D. and Zuckerman, G., Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), 51–90.Google Scholar