Message to Peter Sarnak

Author's comments (Apr. 6, 2014): The concept ``Langlands program'' appears in the title of an article by Stephen Gelbart in the BAMS of April, 1984, but Gelbart himself assured me that it was already current, at least orally, before then. He also drew my attention to a phrase of Armand Borel in his Bourbaki seminar of June, 1975, ``plutôt un vaste programme, élaboré par R. P. Langlands depuis environ 1967.'' I do not recall that I was uneasy with the phrase ``Langlands program'' in 1984, but it then referred principally to matters on which I myself had long reflected. This has since changed. It has come to refer to a domain much larger than the analytic theory of automorphic forms and its arithmetic applications.

This is reflected in the message to Sarnak. Although this perhaps does not correspond exactly to the historical development, the enlargement can be described in two stages. I discuss them separately. The first is the extension of the theory---both the established and, in some regards, also the conjectural form---from a finite extension \(F\) of \(\mathbf Q\) as the base field to a different kind of base field, to an algebraic function field \(F\) over an algebraic curve associated to a field of constants that is either a finite field or the complex number field \(\mathbf C\). It is the second possibility that has the novel ramifications. My source of information on the first and on the second enlargement has been various articles of Edward Frenkel. These articles are impressive achievements but often freewheeling, so that, although I have studied them with considerable care and learned a great deal from them that I might never have learned from other sources, I find them in a number of respects incomplete or unsatisfactory.

As I attempted to explain in the Mostow lecture, I believe that the geometric theory as such should be separated from any kind of duality in physics and treated purely mathematically with less emphasis on sheaf theory than is usually met and with a larger dose of ``classical mathematics'': spectral theory, differential geometry, and algebraic geometry. The algebraic geometry and the spectral theory will, I believe, have to mesh, thus one will be obliged, despite the algebraic geometers, to work with the full classifying space \(\mathrm{Bun}_G\). I am hopeful that such a theory can be constructed in a satisfactory and natural fashion, but this demands a mastery of the pertinent mathematics.

It is the third item of the message to Sarnak whose complexity I did not adequately appreciate as I wrote it. Before explaining this, let me comment on the structures implicit in the classical theory and in the geometric theory. In the classical theory, there is a reductive group \(G\) over a number field and a reductive group \({}^LG\) over \(\mathbf C\). The second group may have several components whose source lies in Galois theory, but that is not the issue here. Their relation is expressed by functoriality. A second aspect of the theory is the relation between the group \({}^LG\) and the Galois group of \(F\) or, more conjecturally, the motivic group over \(F\). This I refer to as reciprocity, the first manifestation of the pertinent phenomena being the law of quadratic reciprocity. Neither aspect has yet been developed to the extent I believe possible, but both principles have led to strong results whose mathematical importance is undeniable and unlikely to be ephemeral.

In the geometric theory as such there is, so far as I can see, no reciprocity, just functoriality and this manifests itself as a parametrization of automorphic representations by connections, in the sense of differential geometry, with values in \({}^LG\), which is often taken to be connected. The possible construction of such a theory, envisioned in the third item of the message, is one of the problems considered in the Mostow lecture. The marked difference between the geometric theory over a closed nonsingular Riemann surface and the arithmetic theory is that in the geometric theory \(G\) and \({}^LG\) are groups of the same type, namely effectively groups over \(\mathbf C\). This leads to a third possibility in addition to functoriality and reciprocity and to a second stage, namely duality. Here we meet problems outside the domain of pure mathematics.

One popular introduction to the topic is Frenkel's Bourbaki lecture, Gauge theory and Langlands duality. On the first page, he describes electro-magnetic duality as an aspect of the Maxwell equations and their quantum-theoretical form or, more generally, as an aspect of four-dimensional gauge theory. This duality is quite different than the functoriality and reciprocity introduced in the arithmetic theory. It entails a supplementary system of differential equations. Moreover, it has to be judged by different criteria. One is whether it is physically relevant. There is, I believe, a good deal of scepticism, which, if I am to believe my informants, is experimentally well-founded. Although the notions of functoriality and reciprocity have, on the whole, been well received by mathematicians, they have had to surmount some entrenched resistance, perhaps still latent. So I, at least, am uneasy about associating them with vulnerable physical notions. On the other hand, as strictly mathematical notions this duality and various attendant constructions, such as the Hitchin fibration, appear to have proven value, especially for topologists and geometers. Whether it is equal to that of functoriality and reciprocity is open to discussion.

In contrast to what I foresaw when describing item (iii), the considerable reflection on duality contemplated there will demand a more sophisticated understanding of topology, geometry, and the relevant physics than I can ever hope to possess.

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