Automorphic induction for is a case of endoscopic transfer, and its character identity was established by Henniart and Herb, up to a constant of proportionality. We determine this constant in terms of the Kottwitz-Shelstad normalization of transfer factors, which involves certain -factors.
"On the Kottwitz-Shelstad normalization of transfer factors for automorphic induction for ." Nagoya Math. J. 208 97 - 144, December 2012. https://doi.org/10.1215/00277630-1815222