Ankita Pal | Searcharxiv

arXiv2025

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

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.

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arXiv2023

Irreducible unitary representations with non-zero relative Lie algebra cohomology of the Lie group $SO_0(2,m)$

By a theorem of D. Wigner, an irreducible unitary representation with non-zero $(\frak{g},K)$-cohomology has trivial infinitesimal character, and hence up to unitary equivalence, these are finite in number. We have determined the number of equivalence classes of these representations and the Poincaré polynomial of cohomologies of these representations for the Lie group $SO_0(2,m)$ for any positive integer $m.$ We have also determined, among these, which are discrete series representations and holomorphic discrete series representations.

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