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Boundary behaviour of special cohomology classes arising from the Weil representation

Part of: Discontinuous groups and automorphic forms Lie groups

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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