Geometric cycles in compact locally Hermitian symmetric spaces and automorphic representations

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let $\theta$ be the Cartan involution of $G$ that fixes $K$. Let $\Lambda$ be a uniform lattice in $G$ such that $\theta(\Lambda)=\Lambda.$ Suppose that $G$ is one of the groups $SU(p,q), p<q-1, q\ge 5, SO_0(2,q)$, $Sp(n,\mathbb{R}), n\ne 4, SO^*(2n), n\ge 9.$ Then there exists a unique irreducible unitary representation $\mathcal{A}_\mathfrak{q}$ associated to a proper $\theta$-stable parabolic subalgebra $\mathfrak{q}$ with $R_+(\mathfrak{q})=R_-(\mathfrak{q})$ such that if $H^{s,s}(\mathfrak{g},K;A_{\mathfrak{q}',K})\ne 0$ for some $0<s\le R_+(\mathfrak{q})$, then $\mathcal{A}_{\mathfrak{q}'}$ is unitarily equivalent to either the trivial representation or to $ \mathcal{A}_{\mathfrak{q}}$. As a consequence, under suitable hypotheses on $\Lambda,$ we show that the multiplicity of $\mathcal{A}_\mathfrak{q}$ occurring in $L^2(\Gamma\backslash G)$ is positive for {\it any} torsionless lattice $\Gamma\subset G$ commensurable with $\Lambda$.