Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $SO_0(2,m)$

TL;DR

This paper investigates special cycles in compact locally Hermitian symmetric spaces of type III related to the Lie group $SO_0(2,m)$, identifying those that contribute non-zero cohomology classes and analyzing their relation to cohomologically induced representations.

Contribution

It classifies special cycles associated with involutions commuting with the Cartan involution and determines which contribute to non-zero cohomology, linking them to specific cohomologically induced representations.

Findings

Identified special cycles that yield non-zero cohomology classes.

Determined the relation between special cycles and cohomologically induced representations.

Analyzed the structure of special cycles in relation to the Lie group $SO_0(2,m)$.

Abstract

Let $G = SO_0(2,m),$ the connected component of the Lie group $SO(2,m);\ K = SO(2) \times SO(m),$ a maximal compact subgroup of $G;$ and $\theta$ be the associated Cartan involution of $G.$ Let $X = G/K,\ \frak{g}_0$ be the Lie algebra of $G$ and $\frak{g} = \frak{g}_0^\mathbb{C}.$ In this article, we have considered the special cycles associated with all possible involutions of $G$ commuting with $\theta.$ We have determined the special cycles which give non-zero cohomology classes in $H^*(\Gamma \backslash X; \mathbb{C})$ for some $\theta$-stable torsion-free arithmetic uniform lattice $\Gamma$ in $G,$ by a result of Millson and Raghunathan. For each cohomologically induced representation $A_\frak{q}$ with trivial infinitesimal character, we have determined the special cycles for which the non-zero cohomology class has no $A_\frak{q}$-component, via Matsushima's isomorphism.