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Boundary behaviour of special cohomology classes arising from the Weil representation.

Funke, Jens and Millson, John (2013) 'Boundary behaviour of special cohomology classes arising from the Weil representation.', Journal of the Institute of Mathematics of Jussieu., 12 (3). pp. 571-634.

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 X ¯ ¯ ¯ of X . However, for the 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 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.

Item Type:Article
Keywords:Weil representation, Cohomology of locally symmetric spaces.
Full text:PDF (arXiv version) - Other (622Kb)
Full text:PDF - Published Version (596Kb)
Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1017/S1474748012000795
Publisher statement:© Copyright Cambridge University Press 2012. This paper has been published in a revised form subsequent to editorial input by Cambridge University Press in "Journal of the Institute of Mathematics of Jussieu" (20012) http://journals.cambridge.org/action/displayJournal?jid=JMJ
Record Created:12 Sep 2012 11:05
Last Modified:07 Jul 2013 00:30

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