Functoriality for the classical groups using the generalized doubling method
. One of the primary goals of number theory is to understand the absolute Galois group of the rational numbers. An important goal of the Langlands Program, is to understand the finite dimensional representations of this group, using automorphic representations. In the other direction, passing from the Galois side to the automorphic setting, this can be regarded as an arithmetic parametrization of local and automorphic representations. This parametrization, which is conjectural in part, predicts a transfer or lifting of local and automorphic representations between two reductive algebraic groups.
The global lift has been established in the celebrated work of Cogdell, Kim, Piatetski-Shapiro and Shahidi for automorphic representations admitting a certain Fourier functional, using the Converse Theorem; and in general by Arthur and by Mok, using the trace formula.
In a joint work with Cai and Friedberg, we present a new proof of functoriality, using integral representations, which generalizes the work of Cogdell et al. to arbitrary automorphic representations. This proof is based on our recent collaboration with Ginzburg, where we generalized the classical doubling method. It is expected to have further applications to the problems of descent and to covering groups.