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 (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 $G'=\mathrm{ U}(m,m)$, $\mathrm{ O}(m,m)$ or $\mathrm{ Sp}(m,m)$ which forms a reductive dual pair with $G$ in the sense of Howe.
Categories: math.GT math.NT