# Geometric cycles in compact locally Hermitian symmetric spaces and automorphic representations

## TL;DR

This paper investigates the relationship between geometric cycles in certain Hermitian symmetric spaces and automorphic representations, establishing conditions under which specific unitary representations appear with positive multiplicity in the space of square-integrable functions on quotients by lattices.

## Contribution

It identifies unique irreducible unitary representations associated with particular parabolic subalgebras and proves their positive multiplicity in automorphic spectra for a class of Lie groups.

## Key findings

- Existence of a unique irreducible unitary representation linked to specific parabolic subalgebras.
- Non-vanishing cohomology implies the presence of these representations in automorphic spectra.
- Positive multiplicity of these representations in $L^2$ spaces for lattices in the specified groups.

## 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 $SU(p,q), p<q-1, q\ge 5, SO_0(2,q)$, $Sp(n,\mathbb{R}), n\ne 4, SO^*(2n), n\ge 9.$ 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 $H^{s,s}(\mathfrak{g},K;A_{\mathfrak{q}',K})\ne 0$ 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 $\Lambda,$ 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 $\Lambda$.