On Simplicity of Poles of Automorphic L-Functions

The purpose of this paper is to prove the simplicity of possible poles of a large class of automorphic L–functions at s=1. The L–functions are all those obtained from the Langlands-Shahidi method in the full generality of every quasisplit connected reductive group. While in [8] we showed that these L–functions when twisted by grossencharacters which are highly ramified at one place, are entire and non–vanishing for Re(s)≥1 under certain local assumptions, little can be said about the location of poles in the interval [0; 2] in any generality. Even the holomorphy at s=2 is not known for any general class of L–functions. We follow the ideas in [6] and [7], namely, if the L-functions have a pole, then certain induced representations should be unitary. Hence the problem of determining poles of global L–functions is reduced to determining unitarity of certain local induced representations.We conclude the paper by including certain special cases of our results which have important number theoretic applications [9,16].