Theta lifts to certain cohomological representations of indefinite orthogonal groups

TL;DR

This paper constructs orthogonal automorphic forms as theta liftings of holomorphic cusp forms, providing detailed Fourier expansion analysis and confirming their square integrability on indefinite orthogonal groups.

Contribution

It introduces a method to explicitly construct and analyze automorphic forms in a specific cohomological representation via theta lifting, with precise Fourier expansion descriptions.

Findings

Explicit construction of automorphic forms in the cohomological representation

Detailed Fourier expansion computations for the liftings

Confirmation of square integrability of the constructed forms

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 $\Pi$, which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms belonging to $\Pi$ 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 $\Pi$. We will also determine the cuspidal supports of the liftings, then confirm the square integrability of those automorphic forms on orthogonal groups.