Rankin-Selberg integrals, the descent method, and Langlands functoriality

In this article I survey the descent method of Ginzburg, Rallis and Soudry and its main applications to the Langlands functorial lift of automorphic, cuspidal, generic representations on a classical group to (appropriate) GL _n, and to establishing a local Langlands reciprocity law for (split) SO_2n+1 (joint work with D. Jiang). The descent method arises when we consider certain residues of special cases of a family of global integrals, attached to pairs of automorphic, cuspidal representations, one on a classical group G and one on GL_n. The last part of this article focuses on the case G = SO_m (split), and the progress made in a joint work with S. Rallis, towards establishing, via the converse theorem, the functorial lift from any automorphic, cuspidal representation on G to GL_2[m/2].