Doubling constructions and tensor product L-functions: Coverings of the symplectic group

In this work, we develop an integral representation for the partial L-function of a pair of genuine irreducible cuspidal automorphic representations, of the m-fold covering of Matsumoto of the symplectic group and of a certain covering group of, with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank- twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Our global results are subject to certain conjectures, but when they are unconditional for all m. One possible future application of this work will be a Shimura-Type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of.