Fourier-Jacobi type spherical functions for discrete series representations of \(\operatorname{Sp}(2,\mathbb{R})\) (Q2752525)

From the MR-review 92k:11052 of \textit{A. Murase} and \textit{T. Sugano} [``Whittaker-Shintani functions on the symplectic group of Fourier-Jacobi type'', Compos. Math. 79, 321-349 (1991; Zbl 0731.11026)] by Daniel Bump: ``It is a paradigm in the theory of automorphic forms that a Rankin-Selberg integral represents an Euler product if it unfolds to a unique model. The classical example is the Whittaker model, which arises in many cases.'' The local results needed for applications to Rankin-Selberg integrals and liftings include a proof of the uniqueness of the model such as in [\textit{J. A. Shalika}, Ann. Math. (2) 100, 171-193 (1974; Zbl 0316.12010)] for the Whittaker case and explicit formulas for the spherical functions attached to this model such as the well-known Shintani-Kato-Casselman-Shalika formula for spherical Whittaker functions. NEWLINENEWLINENEWLINEIn the paper under review, the author considers the Fourier-Jacobi model and establishes both results for discrete series representations of \(G= Sp(2,\mathbb{R})\), the group of \(4\times 4\) symplectic real matrices. In particular, the author shows that the space of spherical Fourier-Jacobi functions with moderate growth associated to a minimal \(K\) type of a discrete series representation of \(G\) is at most one dimensional. The author also provides explicit, case by case formulas for these spherical functions in terms of Meijer functions. NEWLINENEWLINENEWLINEAnalogous results for \(Sp(n,F)\) where \(F\) is a \(p\)-adic field and the representation is unramified were obtained by \textit{A. Murase} and \textit{T. Sugano} (uniqueness), [loc. cit.] and by \textit{A. Murase} (explicit formulas) [Abh. Math. Semin. Univ. Hamb. 61, 153-162 (1991; Zbl 0757.22003)]. A uniqueness result for all representations of \(Sp(2,F)\) where \(F\) is a \(p\)-adic field was proved by the reviewer and \textit{S. Rallis} [J. Lond. Math. Soc. (2) 62, 183-197 (2000; Zbl 0953.22019)], and a similar result is expected in the Archimedean case.