Robert P. Langlands

Editorial comments: The conjectures made in the 1967 letter to Weil were explained here more fully. This appeared originally as a Yale University preprint, later in the published proceedings of a conference in Washington, D.C. Lectures in modern analysis and applications III, Lecture Notes in Mathematics 170, Springer-Verlag, 1970. The lecture is dedicated to Salomon Bochner. 

Author's comments: Although the principal purpose of this paper was to review how the Selberg trace formula is combined with character formulas to calculate the dimension of various spaces of automorphic forms, it is included as a paper on representation theory because the most influential observation in the paper was the description of a possible realization of the discrete series representations on spaces of $L^2$-cohomology.

Author's comments: This paper, or some aspects of this paper, have been called into question in

https://mathoverflow.net/q/144419.

Editorial comments: The letter to Weil included a number of striking conjectures which eventually changed much of the direction of research in automorphic forms. Some of their consequences were explained in a graduate course given at Princeton in the spring of 1967, and then things were put in a somewhat wider context in a series of lectures at Yale later that Spring. These notes were previously published as the first of the Yale Mathematical Monographs.

Editorial comments: This has not been published before. It was written around 1977, just after A. Knapp and G. Zuckerman had announced their results on reducible unitary principal series, subsequently explained in a talk at the A.M.S. 1977 summer school in Corvallis (pp. 93--105 of the published proceedings of that conference.)

Editorial comments:

Editorial comments: In January of 1967, while he was at Princeton University, Langlands wrote a letter of 17 hand-written pages to Andre Weil outlining what quickly became known as `the Langlands conjectures'. This letter even today is worth reading carefully, although its notation is by present standards somewhat clumsy. It was in this letter that what later became known as the `$L$-group' first made its appearance, like Gargantua, surprisingly mature.