Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $S{O}_0(2,m)$

Let $G=S{O}_0(2,m),$ the connected component of the Lie group $SO(2,m);K=SO(2)\times SO(m),$ a maximal compact subgroup of $G;$ and $\theta$ be the associated Cartan involution of $G\mathrm{.}$ Let $X=G\mathrm{/}K,{\mathfrak{g}}_0$ be the Lie algebra of $G$ and $\mathfrak{g}={\mathfrak{g}}_0^{\mathbb{C}}\mathrm{.}$ In this article, we have considered the special cycles associated with all possible involutions of $G$ commuting with $\theta \mathrm{.}$ We have determined the special cycles which give non-zero cohomology classes in $H^{\ast }(\mathrm{\Gamma }\mathrm{\backslash }X;\mathbb{C})$ for some $\theta$-stable torsion-free arithmetic uniform lattice $\mathrm{\Gamma }$ in $G,$ by a result of Millson and Raghunathan. For each cohomologically induced representation $A_{\mathfrak{q}}$ with trivial infinitesimal character, we have determined the special cycles for which the non-zero cohomology class has no $A_{\mathfrak{q}}$-component, via Matsushima’s isomorphism.