Let $X$ denote the non-compact globally Hermitian symmetric space of type $DIII$, namely, $\text{ SO}(n,\mathbb{ H})/\text{ U}(n)$. Let $\Lambda$ be a uniform torsionless lattice in $\text{ SO}(n,\mathbb{ H})$. In this note we construct certain complex analytic submanifolds in the locally symmetric space $X_\Gamma:=\Gamma\backslash \text{ SO}(n,\mathbb{ H})/\text{ U}(n)$ for certain finite index sub lattices $\Gamma\subset \Lambda$ and show that their dual cohomology classes in $H^*(X_\Gamma;\mathbb{ C})$ are not in the image of the Matsushima homomorphism $H^*(X_u; \mathbb{ C})\to H^*(X_\Gamma;\mathbb{ C})$, where $X_u=\text{ SO}(2n)/\text{ U}(n)$ is the compact dual of $X$. These submanifold arise as sub-locally symmetric spaces which are totally geodesic, and, when $\Lambda$ satisfies certain additional conditions, they are non-vanishing `special cycles'. Using the fact that $X_\Lambda$ is a K\"ahler manifold, we deduce the occurrence in $L^2(\Lambda\backslash \text{ SO}(n,\mathbb{ H})$ of a certain irreducible representation $(\mathcal{ A}_\mathfrak{ q}, A_\mathfrak{ q})$ with non-zero multiplicity when $n\ge 9$. The representation $\mathcal{ A}_\mathfrak{ q}$ is associated to a certain $\theta$-stable parabolic subalgebra $\mathfrak{ q}$ of $\mathfrak{ g}_0:=\mathfrak{ so}(n,\mathbb{ H})$. Denoting the smooth $\text{ U}(n)$-finite vectors of $A_{ \mathfrak{ q}}$ by $A_{ \mathfrak{ q},\text{ U}(n)}$, the representation $\mathcal{ A}_\mathfrak{ q}$ is characterised by the property that $H^{ p,p}(\mathfrak{ g}_0\otimes\mathbb{ C},\text{ U}(n); A_{ \mathfrak{ q},\text{ U}(n)})\cong H^{ p-n+2,p-n+2}(\text{ SO}(2n-2)/\text{ U}(n-1);\mathbb{ C}),~p\ge 0$, for $n\ge 9$.
Categories: math.RT math.NT
