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Boundary behaviour of special cohomology classes arising from the Weil representation

Funke, Jens; Millson, John

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


Authors

John Millson



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.

Citation

Funke, J., & Millson, J. (2013). Boundary behaviour of special cohomology classes arising from the Weil representation. Journal of the Institute of Mathematics of Jussieu, 12(3), 571-634. https://doi.org/10.1017/s1474748012000795

Journal Article Type Article
Acceptance Date Mar 1, 2012
Online Publication Date Jul 3, 2012
Publication Date Jul 1, 2013
Deposit Date Mar 19, 2012
Publicly Available Date Sep 14, 2012
Journal Journal of the Institute of Mathematics of Jussieu
Print ISSN 1474-7480
Electronic ISSN 1475-3030
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 12
Issue 3
Pages 571-634
DOI https://doi.org/10.1017/s1474748012000795
Keywords Weil representation, Cohomology of locally symmetric spaces.

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