Geometric cycles in compact locally Hermitian symmetric spaces and automorphic representations

Abstract

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 $SU(p,q),p<q-1,q\ge 5,S{O}_0(2,q)$, $Sp(n,\mathbb{R}),n\ne 4,S{O}^{\ast }(2n),n\ge 9.$ 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 $H^{s,s}(\mathfrak{g},K;A_{{\mathfrak{q}}^{\prime },K})\ne 0$ for some $0<s\le R_{+}(\mathfrak{q})$, then ${\mathcal{A}}_{{\mathfrak{q}}^{\prime }}$ 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 $\Lambda$.

34 pages, 4 figures