Components of harmonic Poincare duals of special cycles

Abstract: The de Rham complex of a compact locally symmetric space \Gamma\ G /K is isometric to the cochain complex C*(g,K; C^\infty(\Gamma\ G)_K) of the relative Lie algebra cohomology of (g,K) with coefficients in C^\infty(\Gamma\ G)_K. This gives an orthogonal decomposition of the space of harmonic forms on \Gamma\ G /K into cochain groups of the form C*(g,K;V), where V is an isotropical sub-represebtation of C^\infty(\Gamma\ G)_K on which the Casimir operator acts trivially. Using representation theoretic methods, we will deduce some conditions for vanishing of a component of the harmonic Poincare dual of a special cycle \Gamma'\ G' /K'$ with respect to this decomposition. For certain special cycles, when  G=SU(p,q), these conditions can be applied to deduce which components do not vanish.