A corollary to Bernstein's theorem and Whittaker functionals on the metaplectic group

Abstract

In this paper, we extend and apply a remarkable theorem due to Bernstein, which was proved in a letter to Piatetski-Shapiro in the fall of 1985 [3]. The theorem is quite useful for establishing analytic continuation and rationality results in a variety of settings in the study of automorphic forms and L-functions, such as in the theory of local intertwining operators and local coefficients for induced representations, or in the theory of Eisenstein series. Bernstein's theorem is extraordinary in terms of the simplicity of its formulation, the elegance of its proof, and its wide range of applicability to problems requiring a proof of analytic continuation. The first section of this paper is devoted to a review of the statement and proof of Bernstein's theorem. We then prove a corollary that can be used in some situations to obtain more precise information about the location of poles. This corollary has already been applied by Friedberg and Goldberg [5] in their work on local coefficients for non-generic representations. In the second section, we apply our corollary to Bernstein's theorem to establish the analytic continuation of the Whittaker functionals for a series of induced representations on the n-fold metaplectic cover of GLr(F), where F is a nonarchimedean local field.