Geometric aspects

The aim of this series of lectures is to illustrate some of the geometric aspects of the theta correspondence. More specifically, we will consider the manifolds defined as quotients Γ\D of the Riemannian symmetric spaces D = D(p,q) for orthogonal groups G=O(p,q) by arithmetic subgroups Γ. Many interesting and important spaces arise in this way, including modular curves, Hilbert modular surfaces, hyperbolic 3-manifolds and others. Theta series for the dual pair (G',G) where G'=Sp(n,R) (or its metaplectic cover), can be used to construct closed differential nq-forms on Γ\D. These forms and their cohomology classes are closely connected to the special cycles in Γ\D, submanifolds arising from subsymmetric spaces D(p',q') in D(p,q). For example, certain generating series for cohomology classes of special cycles can be shown to be modular forms, when n=1, or Siegel modular forms, for n>1.