Hecke $L$-functions and Fourier coefficients of covering Eisenstein series

Abstract

We consider in this paper covering groups and Fourier coefficients of Eisenstein series for induced representations from certain distinguished theta representations. It is shown that one has global factorization of such Fourier coefficients, and the local unramified Whittaker function at the identity can be computed from the local scattering matrices. For a special family of covering groups of the general linear groups, we show that the Fourier coefficients of such Eisenstein series are reciprocals of Hecke $L$-functions, which recovers an earlier result by Suzuki for Kazhdan-Patterson covering groups. We also consider covers of the symplectic group and carry out a detailed analysis in the rank-two case.