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 {O}}}^*(2n) . These are algebraic covered by subgroups of G which same type. show that Poincaré duals these can be viewed as Fourier coefficients a theta series. This gives new cases lifts from cohomology manifolds vector-valued automorphic functions $$G'={{\mathrm {U}}}(m,m)$$ {O}}}(m,m)$$ or {Sp}}}(m,m)$$ forms reductive dual pair with G.