Applications of Langlands' functorial lift of odd orthogonal groups (Q2782663)

Langlands' functorial lift is one of the central questions in the theory of automorphic forms. The existence of a weak functorial lift of a generic cuspidal representations of \(SO_{2n+1}\) to \(GL_{2n}\) was proved by \textit{J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi} [Publ. Math., Inst. Hautes Étud. Sci. 93, 5--30 (2001; Zbl 1028.11029)]. An explicit characterization of the image of the Langlands functorial lift for odd-orthogonal groups is due to \textit{D. Ginzburg, S. Rallis} and \textit{D. Soudry} [Int. Math. Res. Not. 2001, No. 14, 729-764 (2001; Zbl 1060.11031)].NEWLINENEWLINEThe paper under review gives a simpler proof in the case that a cuspidal representation has one supercuspidal component. The paper also gives several applications of the functorial lift to local problems: First, a parametrization of square-integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadić. Second, a criterion for cuspidal reducibility of supercuspidal representations of GL\(_m \times \text{SO}_{2n+1}\). Third, a functorial lift from generic cuspidal representations of \(\text{SO}_5\) to automorphic representations of \(GL_5\), corresponding to the \(L\)-groups homomorphism \(\text{Sp}_4({\mathbb C}) \to \text{GL}_5({\mathbb C})\), given by the second fundamental weight.