Cohomology of compact locally symmetric spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S(Gamma)=Gamma \X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup Gamma of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S-H(Gamma)=Delta \Y of Gamma \X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in C-n and C-m, then low degree cohomology classes on the variety S(Gamma) restrict non-trivially to the subvariety S-H(Gamma); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S(Gamma).