Rankin–Selberg Integrals and L-Functions for Covering Groups of General Linear Groups

Abstract

Abstract Let ${\operatorname {GL}}_c^{(m)}$ be the covering group of ${\operatorname {GL}}_c$, obtained by restriction from the $m$-fold central extension of Matsumoto of the symplectic group. We introduce a new family of Rankin–Selberg integrals for representations of ${\operatorname {GL}}_c^{(m)}\times {\operatorname {GL}}_k^{(m)}$. The construction is based on certain assumptions, which we prove here for $k=1$. Using the integrals, we define local $\gamma $-, $L$-, and $\epsilon $-factors. Globally, our construction is strong in the sense that the integrals are truly Eulerian. This enables us to define the completed $L$-function for cuspidal representations and prove its standard functional equation.