2011-02-11

Mahdi Asgari, Oklahoma State University

Langlands Functoriality for General Spin Groups

Let G_1 and G_2 be a pair of connected reductive linear algebraic groups. The Functoriality Principle, conjectured by Robert Langlands in the sixties, roughly asserts that, associated with a map from the L-group of G_1 to the L-group of G_2, one should be able to transfer automorphic representations from G_1 to G_2 preserving the corresponding automorphic L-functions. Given that automorphic representations are quite rigid objects and encode a lot of data, functoriality turns out to encompass some very important number-theoretic statements, among them the Artin conjecture, the generalized Ramanujan conjecture, and the Selberg eigenvalue conjecture. I will report on a project, joint with F. Shahidi, we started a few years ago and completed recently that establishes Langlands Functoriality from the quasi-split general spin groups (type B_n or D_n) to the general linear group GL(2n) associated with a natural embedding of their L-groups to that of GL(2n) for a rather large family of automorphic representations, called (globally) generic. We established a weak form of this result for the split cases a few years ago, but were recently able to complete that result to cover the quasi-split non-split case, describe the image of this transfer completely and give a number of applications. The analogous result for special orthogonal and symplectic groups is due to Cogdell, Kim, Piatetski-Shapiro and Shahidi alogn with Ginzburg, Rallis, and Soudry building on the earlier works of Gelbart and Piatetski-Shapiro.