The genesis and gestation of functoriality

Robert P. Langlands
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Author's comments: The text on the genesis and gestation of functoriality was for an informal lecture at the Tata Institute of Fundamental Research in Bombay delivered on short notice at the suggestion of Venkataraman. It has been suggested to me that the first four pages, a brief summary of the development of the theory of automorphic forms before 1960, roughly as it affected my initial reflections, manage to be simultaneously trite and eccentric and might be best omitted. The reader is free to do so, but the purpose of posting this text is to record the lecture, not to improve it.

Although I attach some importance to the historical origins of the theory, even to my own understanding of them, and am not entirely persuaded that all contemporary readers will be fully aware of the mathematical climate in the early 1960's or of the various strands in the number theory of the nineteenth and twentieth century, it may be best to consider the doubtful first four pages simply as remarks in the way of warm-up.

On the other hand, in retrospect, one perhaps curious feature of my mathematical education and development is that I never studied elementary number theory, either on my own or formally, and some obvious things never appealed to me. Although I mention Weyl's book on algebraic number theory in the lecture, I do not confess that it was only on reading it that I began to appreciate the beauty of the law of quadratic reciprocity to which I had earlier attached no importance. I like now to think that I would have greatly benefited from an introduction to Gauss's Disquisitiones by a perceptive teacher at the beginning of my career. So I may, after all, having been trying to convey autobiographical information even in the warm-up.

There is a specific point, perhaps not entirely devoid of interest, that did not occur to me at the time of the Mumbai lecture. It can be more or less squared with the recollections there of lectures at Yale in 1967. At Yale, I listed on the blackboard the representations \(\rho\) of the \(L\)-group for which \(L(s,\pi,\rho) \) could be analytically continued by what is now known as the Langlands-Shahidi method. I had calculated them case by case. It was Jacques Tits who immediately observed that these were the representations on the unipotent radical of the Levi subgroup of the dual group associated to the Levi subgroup of \(G\) defining the pertinent Eisenstein series. So I was perhaps not so reticent at Yale as I suggested in Mumbai.