Theta series and generalized special cycles on Hermitian locally symmetric manifolds

We study generalized special cycles on Hermitian locally symmetric spaces $\mathrm{\Gamma }\mathrm{\backslash }D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{S}\mathrm{p}(2n,\mathbb{R})$ and ${\mathrm{O}}^{\ast }(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator representation and a construction which essentially comes from the thesis of Greg Anderson, we show that Poincar'e 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 Hermtian locally symmetric manifolds associated to $G$ to vector valued automorphic forms associated to the groups $G^{\mathrm{\prime }}=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{S}\mathrm{p}(m,m)$ which forms a reductive dual pair with $G$ in the sense of Howe.