The Langlands-Shahidi L-Functions for Brylinski-Deligne Extensions

We firstly discuss properties of the $L$-groups for Brylinski-Deligne extensions of split reductive groups constructed by M. Weissman. Secondly, the Gindikin-Karpelevich formula for an arbitrary Brylinski-Deligne extension is computed and expressed in terms of naturally defined elements of the group. Following this, we show that the Gindikin-Karpelevich formula can be interpreted as Langlands-Shahidi type $L$-functions associated with the adjoint action of the $L$-group for the Levi covering subgroup on certain Lie algebras. As a consequence, the constant term of Eisenstein series for Brylinski-Deligne extensions could be expressed in terms of global (partial) Langlands-Shahidi type $L$-functions. These $L$-functions are shown to possess meromorphic continuation to the whole complex plane. In the end, we determine the residual spectra of Brylinski-Deligne covers of some semisimple rank one groups.