Theta lifts to certain cohomological representations of indefinite orthogonal groups

Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O(b⁺,b^-) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π, which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp₂(R) cusp forms by using the Borcherds’ method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions.