Theta series and generalized special cycles on Hermitian locally symmetric manifolds

Abstract

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 ${\mathrm{O}}^{\ast }(2n)$. 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 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 Hermtian locally symmetric manifolds associated to $G$ to vector valued automorphic forms associated to the groups $G^{\prime }=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{Sp}(m,m)$ which forms a reductive dual pair with $G$ in the sense of Howe.

References in corpus (2)

• The Geometric Theta Correspondence for Hilbert Modular Surfaces
• Produits dans la cohomologie des variétés arithmétiques : quelques calculs sur les séries thêta