Boundary Behavior of Special Cohomology Classes Arising from the Weil Representation

In our previous paper [10], we established a theta correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces attached to real orthogonal groups. This correspondence is realized via theta functions associated to explicitly constructed ”special” Schwartz forms. These forms give rise to relative Lie algebra cocycles for the orthogonal group with values in the Weil representation tensored with the coefficients. In this paper, we systematically study certain complexes associated to the Weil representation and introduce a ”local” restriction map for the Weil representation to a face of the Borel-Serre enlargement of the underlying orthogonal symmetric space. We then compute the restriction of the special cocycles. As a consequence we obtain a formula for the (global) restriction of the associated theta function to the Borel-Serre boundary of orthogonal local symmetric spaces. The global restriction is again a theta series as in [10] for a smaller orthogonal group and a larger coefficient system.