Doubling Constructions: the complete L-function for coverings of the symplectic group

Abstract

We develop the local theory of the generalized doubling method for the m-fold central extension Sp (m) 2n of Matsumoto of the symplectic group. We define local γ-, L-and ǫ-factors for pairs of genuine representations of Sp (m)2n × GL k and prove their fundamental properties, in the sense of Shahidi. Here GL k is the central extension of GL k arising in the context of the Langlands-Shahidi method for covering groups of Sp 2n × GL k . We then construct the complete L-function for cuspidal representations and prove its global functional equation. Possible applications include classification results and a Shimura type lift of representations from covering groups to general linear groups (a global lift is sketched here for m = 2).