A Bessel identity for the theta correspondence
In this paper, the authors prove certain global identity between Bessel distributions with respect to generic cuspidal representations of an odd orthogonal group and the metaplectic cover of a symplectic group (of the same semisimple rank as this orthogonal group), which are related through the theta correspondence. They also prove the local counterpart of this result (and actually this calculation comprises the bulk of the paper), in the case of square integrable generic representations of the groups in question. Using some previous results of (some of the) authors, and other people (older results of Casselman, etc.) -- mainly about estimates of the values of Whittaker functions on split maximal torus -- they explicitly realize a generic, square-integrable representation in the space of Whittaker functions. They prove that this realization is actually in the space of square-integrable Whittaker functions (cf. section 2). Using these realizations, they explicitly realize generic theta correspondence in the equal rank-case -- as the bijection given by an integral transform -- formally very similar to the global theta integral (in the case of square-integrable representations). Then, using these results, and introducing Bessel distributions, they also prove the identity of Bessel distributions for matching test-functions (i.e., with compatible orbital integrals). In the last part of the paper, the authors prove analogous global results; first they prove that theta kernels give bijection between those generic cuspidal automorphic representations (on both global groups) for which the theta lift is non-zero. They also characterize which are the representations which have the non-zero theta lifts. Also this prepares the final global result, which is an appropriate analog of the matching in the local case.
scientific article; zbMATH DE number 6176544
scientific article