Stability of γ-factors for quasi-split groups

One of the main obstacles in applying converse theorems to prove new cases of functonality is that of stability of gamma-factors for a certain class of L-functions obtained from the 'Langlands-Shahidi' method, where the gamma-factors are defined inductively by means of 'local coefficients'. The problem then becomes that of stability of local coefficients upon twisting the representation by a highly ramified character. In this paper we first establish that the inverses of certain local coefficients are, up to an abelian gamma-factor, genuine Mellin transforms of partial Bessel functions of the type we analysed in our previous paper. The second main result is then the resulting stability of the local coefficients in this situation, which include all the cases of interest for functoriality. Hopefully, the analysis given here will open the door to a proof of the general stability and the equality of gamma-factors obtained from different methods through integration over certain quotient spaces whose generic fibres are closed. They do not seem to have been studied before in any generality.