Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $SO_0(2,m)$
Ankita Pal, Pampa Paul
TL;DR
This work analyzes special geometric cycles in compact locally Hermitian symmetric spaces associated with G = SO_0(2,m) by exploiting involutions σ commuting with the Cartan involution to produce fixed-point cycles. It couples Millson–Raghunathan’s results on non-zero cohomology classes with Matsushima’s isomorphism to connect these cycles to automorphic representations, and it systematically identifies, for each A_q with trivial infinitesimal character, which cycles contribute or do not contribute to the cohomology. The authors construct cocompact arithmetic lattices over a totally real field, classify the relevant involutions (via Vogan diagrams), and determine orientation-preserving properties of the corresponding fixed-point subgroups. By integrating these geometric cycles with the representation-theoretic framework of cohomologically induced representations, the paper provides a concrete bridge between the geometry of special cycles and the automorphic spectrum for G = 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 $θ$ 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 $θ.$ We have determined the special cycles which give non-zero cohomology classes in $H^*(Γ\backslash X; \mathbb{C})$ for some $θ$-stable torsion-free arithmetic uniform lattice $Γ$ 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.