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) $ defined by Millson and the author in their joint work. 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 $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n) $ to vector valued automorphic forms associated to the groups $G'=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{Sp}(m,m)$ which are dual groups of $G$ in the sense of Howe. The above three groups are all the groups that show up in real reductive dual pairs of type I in the sense of Howe whose symmetric space is of Hermitian type with the exception of $\mathrm{O}(p,2)$. In a following paper, Millson and the author will construct theta series for the rest of the groups that show up in real reductive dual pairs of type I.