Applications of Langlands’ Functorial Lift of Odd Orthogonal Groups

Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of SO2n+1 to GL2n. Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of GLni (more precisely, cuspidal representations of GL2ni such that the exterior square L-functions have a pole at s = 1). One purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component. In a separate paper, we prove it without any condition using a result on spherical unitary dual due to Barbasch and Moy. We give several applications of the functorial lift: First, we parametrize square integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadic. Second, we give a criterion for cuspidal reducibility of supercuspidal representations of GLm × SO2n+1. Third, we obtain a functorial lift from generic cuspidal representations of SO5 to automorphic representations of GL5, corresponding to the L-group homomorphism Sp4(C) −→ GL5(C), given by the second fundamental weight.