Local coefficients as Mellin transforms of Bessel functions: towards a general stability

The main obstacle in extending the breakthrough in transferring generic cusp forms on odd special orthogonal groups to automorphic forms on general linear groups, recently established by the author and his collaborators, is the stability of corresponding root numbers under highly ramified twists. In this paper, the author proves a general result, expressing the reciprocals of his local coefficients for an arbitrary quasisplit group, basically as Mellin transforms of certain incomplete Bessel functions. Local root numbers are then defined inductively, using local coefficients. When combined with ideas of Cogdell and Piatetski-Shapiro on Bessel functions, this leads to a proof of stability in a number of new cases, including all the classical groups.