Abstract
Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups \(\textrm{O}({b^+},{b^-})\) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule \(\Pi \), which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to \(\Pi \) as theta liftings of holomorphic \(\textrm{Mp}_2({\mathbb {R}})\) 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.
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Miyazaki, T., Saito, Y. Theta lifts to certain cohomological representations of indefinite orthogonal groups. Res. number theory 10, 25 (2024). https://doi.org/10.1007/s40993-024-00510-z
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DOI: https://doi.org/10.1007/s40993-024-00510-z