# Theta series and generalized special cycles on Hermitian locally symmetric manifolds

## TL;DR

This paper links generalized special cycles on Hermitian symmetric spaces to theta series, enabling new theta lifts from cohomology to automorphic forms for certain dual pairs, advancing understanding of automorphic representations.

## Contribution

It establishes a novel connection between special cycles and theta series on Hermitian symmetric spaces, expanding the scope of theta lifts in automorphic representation theory.

## Key findings

- Poincaré duals of special cycles are Fourier coefficients of theta series.
- New cases of theta lifts from cohomology to automorphic forms are identified.
- The work applies oscillator representations and dual pairs in Howe's theory.

## 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 $\mathrm{O}^*(2n)$. 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.