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 ⊂ G be a maximal compact subgroup of G. Suppose that the centre of K is isomorphic to 𝕊1 so that X := G/K is a global Hermitian symmetric space. Denote the compact dual of X by Xu. Let θ be the Cartan involution of G that fixes K. Let Γ be a uniform arithmetic lattice in G such that θ(Γ) = Γ. Suppose that G is one of the groups SU(p, q), p < q – 1, q ≥ 5, SO0(2, q), Sp(n, ℝ), n ≠ 4, SO*(2n), n ≥ 9. Then there exists a unique irreducible unitary representation $$ {\mathcal{A}}_{\mathfrak{q}} $$ associated to a proper θ-stable parabolic subalgebra $$ \mathfrak{q} $$ with R+( $$ \mathfrak{q} $$ ) = R−( $$ \mathfrak{q} $$ ) such that if Hs,s( $$ \mathfrak{g} $$ , K; $$ {A}_{\mathfrak{q}\prime, K} $$ ) ≠ 0 for some 0 < s ≤ 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, using certain geometric cycles C ⊂ XΓ := Γ\X constructed by Millson and Raghunathan, we show that under suitable hypotheses on Γ, for any sublattice Λ ⊂ Γ, the representation $$ {\mathcal{A}}_{\mathfrak{q}} $$ occurs in L2(Λ\G) with positive multiplicity.