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 $\pi\times\tau$ of genuine irreducible cuspidal automorphic representations, $\pi$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $\tau$ of a certain covering group of $GL_k$, 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-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Possible applications include an analytic definition of local factors for representations of covering groups, and a Shimura type lift of representations from covering groups to general linear groups.