Applications of Langlands’ functorial lift of odd orthogonal groups

Abstract

Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of SO 2n+1 to GL 2n . 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 GLn i (more precisely, cuspidal representations of GL2n i 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 GL m x SO 2n+1 . Third, we obtain a functorial lift from generic cuspidal representations of SO 5 to automorphic representations of GL 5 , corresponding to the L-group homomorphism Sp 4 (C) → GL 5 (C), given by the second fundamental weight.