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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. http://resolver.caltech.edu/CaltechTHESIS:05272016-115010946

Abstract

We prove a globalization theorem for self-dual representations of GLN over a totally real number field F, which gives a positive existence criterion for self-dual cuspidal automorphic representations of GLN(AF) 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 GLN 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 GLN 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 GLN 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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics, Lie Groups, Langlands program, Automorphic Forms, Automorphic Representations, Trace Formula
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Flach, Matthias
  • Graber, Thomas B.
  • Mantovan, Elena
Defense Date:9 May 2016
Record Number:CaltechTHESIS:05272016-115010946
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:05272016-115010946
DOI:10.7907/Z9WD3XKB
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9788
Collection:CaltechTHESIS
Deposited By: Brian Hwang
Deposited On:09 Mar 2017 17:14
Last Modified:13 Sep 2017 18:43

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