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

Abstract

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 $G.$ Let $X=G/K,{\mathfrak{g}}_0$ be the Lie algebra of $G$ and $\mathfrak{g}={\mathfrak{g}}_0^{\mathbb{C}}.$ In this article, we have considered the special cycles associated with all possible involutions of $G$ commuting with $\theta .$ We have determined the special cycles which give non-zero cohomology classes in $H^{\ast }(\Gamma \backslash X;\mathbb{C})$ for some $\theta$-stable torsion-free arithmetic uniform lattice $\Gamma$ 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.

26 pages, 1 figure, 5 tables