Let G be a unitary, symplectic or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a converse theorem representations.

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 repre...