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.
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