Non-vanishing cohomology classes in uniform lattices of $\text{SO}(n,\mathbb{H})$ and automorphic representations

Abstract

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 $n\ge 9$.