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 $G\mathrm{/}K$ is a global Hermitian symmetric space. Let $\theta$ be the Cartan involution of $G$ that fixes $K$. Let $\mathrm{\Lambda }$ be a uniform lattice in $G$ such that $\theta (\mathrm{\Lambda })=\mathrm{\Lambda }\mathrm{.}$ Suppose that $G$ is one of the groups $SU(p,q),p<q-1,q\ge 5,S{O}_0(2,q)$, $Sp(n,\mathbb{R}),n\mathrm{\ne }4,S{O}^{\ast }(2n),n\ge 9.$ 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 $H^{s,s}(\mathfrak{g},K;A_{{\mathfrak{q}}^{\mathrm{\prime }},K})\mathrm{\ne }0$ 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 $\mathrm{\Lambda },$ we show that the multiplicity of ${\mathcal{A}}_{\mathfrak{q}}$ occurring in $L^2(\Gamma\backslash G)$ispositiveforanytorsionlesslattice$\Gamma\subset G$commensurablewith$\Lambda$.