Special representations of rank one orthogonal groups
Let G be the special orthogonal group of an m-dimensional quadratic space X with Witt index one over a number field k. The authors study (for \(m\geq 5)\) the image of the theta correspondence from \(\overline{Sl}_ 2({\mathbb{A}})\) to G(\({\mathbb{A}})\). The main result of the paper is a characterization of the image of this theta correspondence in terms of certain invariance properties of the Fourier coefficients of the cusp forms in this image. The cusp forms with this invariance property are called special. The proof of the fact that all special cusp forms occur in the image, requires first an analogous local statement (almost everywhere) and then the use of certain Rankin-Selberg integrals. A similar problem for the group PGSp(4)\(\simeq SO(3,2)\) was previously studied by the first-named author in [Prog. Math. 35, 309-325 (1983; Zbl 0566.10021)].
scientific article; zbMATH DE number 4125479
scientific article