Some structural results for the stability of Root Numbers

In the Langlands-Shahidi method, one uses local coefficients to define local root numbers. One needs to prove the stability of these root numbers upon twists by highly ramified characters, in order to establish functoriality using converse theorems. This stability has been established for all cases of interest in functoriality under an assumption that two sets appearing in the calculation of local
coefficients are the same. In this talk, we will discuss my proof of this assumption.
More precisely, let G be a quasi-split connected reductive algebraic group defined   over a local field k. Let B = TU be a Borel subgroup of G over k, and P =MN the Levi decomposition of a standard self associate maximal parabolic P of G over k, where ..............(contd.)