Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group SO₀(2,m)

Let G = SO₀(2,m), the connected component of the Lie group SO(2,m);\ K = SO(2) × SO(m), a maximal compact subgroup of $G;$ and θ be the associated Cartan involution of $G.$ Let X = G/K,\ g₀ be the Lie algebra of G and g = g₀C. In this article, we have considered the special cycles associated with all possible involutions of G commuting with θ. We have determined the special cycles which give non-zero cohomology classes in H^*(Γ X; C) for some θ-stable torsion-free arithmetic uniform lattice Γ in $G,$ by a result of Millson and Raghunathan. For each cohomologically induced representation Aq with trivial infinitesimal character, we have determined the special cycles for which the non-zero cohomology class has no Aq-component, via Matsushima's isomorphism.