Theta series and generalized special cycles on Hermitian locally symmetric manifolds

We study generalized special cycles on Hermitian locally symmetric spaces $$\Gamma \backslash D$$ associated to the groups $$G={{\mathrm {U}}}(p,q)$$ , $${{\mathrm {Sp}}}(2n,{\mathbb {R}}) $$ and $${{\mathrm {O}}}^*(2n) $$ . These cycles are algebraic and covered by symmetric spaces associated to subgroups of G which are of the same type. We show that Poincaré duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to G to vector-valued automorphic functions on the groups $$G'={{\mathrm {U}}}(m,m)$$ , $${{\mathrm {O}}}(m,m)$$ or $${{\mathrm {Sp}}}(m,m)$$ which forms a reductive dual pair with G.