Functoriality for the classical groups
Recently, in collaboration with Kim, Piatetski-Shapiro, and Shahidi, we have established global functoriality from the split orthogonal groups and symplectic groups to GL(N). Supposedly, one of the consequences of functoriality is the ability to then ``pull back'' various structural facts about automorphic representations of GL(N) to these groups. In this lecture I would like to explain how functoriality combined with the descent theory of Ginzburg, Rallis, and Soudry let us carry this out and obtain such global results as Ramanujan type bounds and rigidity theorems for globally generic cuspidal representations as well local results such as extending local functoriality and the local Langlands conjecture to all generic representations of SO(2n+1) over a p-adic field.