On the Langlands Program

The following sections are included:

• Contents
• Foreword
• Preface

This chapter is aimed to give the reader a working understanding of the terms involved in the Arthur-Selberg trace formula for the group SL(2) in the adelic setting closely following the excellent notes [Art05] by Prof. Arthur in the Clay proceedings. We analyze the truncated kernel and develop the coarse and fine geometric and spectral expansions for SL(2). The invariant trace formula is also computed explicitly for SL(2). We end with a brief note on recent convergence results.

This chapter represents an expanded version of the material of five introductory lectures on the theory of endoscopy delivered at the special program “On the Langlands Program: Endoscopy and Beyond” held at the Institute for Mathematical Sciences of the National University of Singapore from 17th December 2018 to 18th January 2019. I wish to thank the organizers of this program, Dihua Jiang, Lei Zhang, Bill Casselman, Pierre-Henri Chaudouard, Wee Teck Gan, and Chengbo Zhu, for their generous hospitality and for the excellent working conditions during the special program. I also thank Ali Altug, Bill Casselman, and Pierre-Henri Chaudouard, for answering various questions related to this chapter.

The purpose of this chapter is to explain Mœglin’s explicit constructions of A-packets both when the base field F is p-adic and when F is archimedean.

The purpose of this chapter is to explain p-adic Adams–Barbasch– Vogan packets constructed by Cunningham, Fiori, Mracek, Moussaoui and Xu [7].

This chapter provides an informal introduction, with examples, to some technical aspects of the re-normalization of measures on orbital integrals used in the work of Langlands, Frenkel-Langlands-Ngô, and Altug on Beyond Endoscopy. In particular, we survey different relevant measures on algebraic tori and explain the connection with the Tamagawa numbers. We work out the example of GL₂ in complete detail. The Appendix by Matthew Koster illustrates, for the Lie algebras 𝔰𝔩₂ and 𝔰𝔬₃, the relation between the so-called geometric measure on the orbits and Kirillov’s measure on co-adjoint orbits in the linear dual of the Lie algebra.

This chapter is based on the six lectures on Beyond Endoscopy delivered at the workshop at the National University of Singapore for the thematic program “On the Langlands Program: Endoscopy and Beyond” in January 2019. The chapter is separated into three parts: the first part is an informal introduction to Beyond Endoscopy following [Lan04]. The second part goes over the content of [Alt15, Alt17, Alt18] emphasizing Poisson summation. In the last part we discus several related works.

This chapter presents a brief introduction to the proposal of A. Braverman and D. Kazhdan ([10]), with additional supplements by B. C. Ngô ([38]).