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

TL;DR

This paper constructs non-vanishing cohomology classes in certain locally symmetric spaces associated with d8(n,\u210b) and demonstrates their connection to automorphic representations, revealing new insights into the cohomology of these spaces.

Contribution

It introduces new complex submanifolds in locally symmetric spaces of d8(n,\u210b), showing their cohomology classes are outside the Matsushima image and linking these to specific automorphic representations.

Findings

Construction of special submanifolds with non-trivial cohomology classes.

Identification of automorphic representations with non-zero multiplicity.

Explicit relation between cohomology and heta-stable parabolic subalgebras.

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 $n\ge 9$.