Constructing Self-Dual Automorphic Representations on General Linear Groups

Citation

Hwang, Brian W. (2016) Constructing Self-Dual Automorphic Representations on General Linear Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9WD3XKB. https://resolver.caltech.edu/CaltechTHESIS:05272016-115010946

Abstract

We prove a globalization theorem for self-dual representations of GL_N over a totally real number field F, which gives a positive existence criterion for self-dual cuspidal automorphic representations of GL_N(A_F) with prescribed local components at a finite set of finite places. A byproduct of our argument is that the automorphic representations that we construct are cohomological (equivalently, regular algebraic) and so fall into the class of automorphic representations on GL_N for which there is a well-established theory for how to attach Galois representations, using the etale cohomology of certain Shimura varieties. The primary motivation is to give a sort of "bare-handed" or "low tech" proof of a result that is implied by the philosophy of twisted endoscopy in the Langlands program. While we are guided by this overarching picture, in the argument itself, we obtain all our results by working directly on GL_N and the group obtained by twisting it under the "inverse-transpose" involution. In particular, we do not appeal to any general results on twisted endoscopic transfer or assume any big "black box" results like the (conjectured) stabilization of the twisted trace formula. Hence, such results are unconditional as stated, and we remark throughout on why the particular assumptions that we impose turn out to be necessary, indicating the (often substantial amount of) additional work required to generalize the stated results.

In an appendix, in stark contrast to our approach above, we give an abridged argument for proving a globalization theorem on GL_N in great generality, assuming a couple of major technical hypotheses (albeit, ones that are widely believed to be true) and yielding to Arthur's endoscopic classiffication of representations of symplectic and special orthogonal groups. Our hope is for such an argument to provide an outline for how we might ultimately prove results like generalizations of the globalization criterion above in the future.

Thesis Files