Theta lifts to certain cohomological representations of indefinite orthogonal groups

Abstract

Howe and Tan (1993) investigated a degenerate principal series representation of indefinite orthogonal groups $\mathrm{O}(V)$ and explicitly described its composition series. They showed that there exists a unique unitarizable irreducible submodule $Π$, which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms belonging to $Π$ as theta liftings of holomorphic $\mathrm{Mp}_2(\mathbb{R})$ cusp forms by using the Borcherds' method. We will give a useful observation in computing their Fourier expansions that enables us to give very precise descriptions of the liftings specially belonging to $Π$. We will also determine the cuspidal supports of the liftings, then confirm the square integrability of those automorphic forms on orthogonal groups.