This is joint work with John.
The Weil representation is a well-known tool to study arithmetic and cohomological
aspects of orthogonal groups.
We construct certain, "special", cohomology classes for orthogonal groups O(p,q) with
coefficients in a finite dimensional representation and discuss their automorphic and
geometric properties. In particular, these classes are generalizations of previous work
of Kudla and Millson and give rise to Poincare dual forms for certain, "special", cycles
with non-trivial coefficients in arithmetic quotients of the associated symmetric space
for the orthogonal group.
Furthermore, we determine the behavior of these classes at the boundary of the
Borel-Serre compactification of the associated locally symmetric space. As a consequence
we are able to obtain new non-vanishing results for the special cycles.