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 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. We have determined the special cycles which give non-zero cohomology classes in H^* (X; C) for some -stable torsion-free arithmetic uniform lattice 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.