Two Messages to Mueller and Volpato

Author's comments: These two letters were, in fact, electronic messages and were written very recently, in December, 2013. They were inspired by a conversation with Julia Mueller and Michael Volpato, in which I tried to explain to them the origins of the general definition of automorphic \(L\)-functions. I had difficulty, not because I had forgotten that these were in the theory of Eisenstein series, but because I had not recognized in 1966, when I discovered after many months of unsuccessful search a promising definition of automorphic \(L\)-function, what a fortunate, although, and this needs to be stressed, unforeseen by me, or for that matter anyone else, blessing it was that it lay in the theory of Eisenstein series. This, I try to explain in the messages. There is something to add to them.

I begin by suggesting that the modern analytic theory of automorphic forms has its origins largely in the work of Hecke and Siegel, and that one decisive step was the extension of the reduction theory to general reductive groups by Borel and Harish-Chandra in Arithmetic subgroups of algebraic groups Ann. Math. (1962). The general \(L^2\)-theory, thus the theory for general reductive groups, especially the notion of cusp form, seems to have had its origins in two papers, one by Godement (in Sém. H. Cartan, 1957/58, Ex-. 8, pp. 8-10) and the other by Harish-Chandra (Automorphic forms on a semisimple Lie group, 1959). The central analytic problem of establishing the spectral theory, continuous and discrete was, on the other hand, broached first, but only in the case of \(\mathrm{SL}(2)\) by Maaß and, through his influence but with stronger results, by Selberg.

Some authors refer to the philosophy of cusp forms introduced by Harish-Chandra but only later, when he was well on his way to the general Plancherel formula, thus some time after he had constructed the discrete series in general and, in particular, after he had the material of the papers Discrete series for semisimple Lie groups, I, II in hand and about the time, 1966/67, that he was lecturing on the theory of Eisenstein series for cusp forms. The papers had appeared in 1965/66; my notes on Eisenstein series earlier. Although I cannot be sure, I believe he undertook the lectures in response, a very generous response, to doubts expressed by various mathematicians about the validity of my paper, which I had delayed publishing in the hope of improving the exposition. So far as I know, he was not moved to express the similarity of the local and global spectral theories as a philosophy until he understood the technical details of both well. It has certainly been of great importance that the local theory, thus the representation theory, and the theory of automorphic representations have developed in tandem since the 1960's and that they have many features in common, but when examined closely, there is also a striking difference: the presence of non-tempered representations in the global theory. The ``philosophy'' as such was appealing when first introduced, and remains so, especially as an expression of Harish-Chandra's personality, but it is not of much technical significance and, although certainly valid for nonarchimedean fields, has not been the key to treating them.

I, myself, came to the modern theory of automorphic forms by reading first Selberg, then Siegel, and then the earlier papers of Harish-Chandra, and for the theory of Eisenstein series in one variable was at first largely concerned with methods with their direct source in papers of Siegel. In particular, I had not noticed the papers of Godement and Harish-Chandra to which I just referred or the importance of the notion of cusp forms for the spectral theory on the quotient \(\Gamma\backslash G\). So the lecture of Gelfand at the 1962 ICM was a revelation to me. It suggested, in recollection at least, almost immediately, a possible way of establishing the analytic continuation of Eisenstein series in general. There would be three steps: (i) the continuation for the rank-one series associated to cusp forms; (ii) the continuation for the series in several variables associated to cusp forms; (iii) the continuation for series in any number of variables associated to forms that are not cuspidal. The first step would be accomplished by the method used by Selberg for discrete subgroups of \(\mathrm{SL}(2)\) or more generally for discrete subgroups of reductive groups of rank one. For the second step, as I had discovered as a graduate student at Yale, standard methods from the theory of functions in several variables that, by a stroke of luck, I came across in my reading for a seminar of Browder/Kakutani on functions of several complex variables, which ultimately did not take place, allow one to pass from groups of rank one to groups of higher rank. I, of course, was initially thinking in very concrete terms. The argument is explained in a somewhat more sophisticated number-theoretical context in the first appendix to the Springer notes (544) on Eisenstein series, included in Part 3 of the present collection. The appendix itself was written I believe during my first one or two years at Princeton and reflects the algebraic number theory that I learned after arriving there. The method was presumably also used by Selberg, but I never discussed this with him.

It was the third step, the construction of the full spectrum from residues of Eisenstein series, closely related to what are now referred to as Arthur packets, that caused me the most difficulty. So far as I know, Gelfand, for example, was not aware of its necessity. Although I began the project that led to the text in the Springer notes not long after reading the Gelfand lecture, thus in the academic year 1962/63, which I had spent at the IAS, I did not finish it until the spring of 1964. I was exhausted. I found the correct induction assumptions for the proof only after many false tries. The next academic year I spent at Berkeley. As I confess in the essay Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele in Part 3, that year was mathematically disappointing. The following academic year, 1965/66 was worse. As I recall in the same essay, I was trying, with no success, to find on the one hand the correct generalization of the Hecke \(L\)-functions to the theory of automorphic forms on general reductive groups and on the other hand some form of a nonabelian class field theory that I could accept as the correct form. I was not succeeding, although about the same time or perhaps a little later, Tamagawa and Godement were beginning to study forms of the standard \(L\)-function associated to \(\mathrm{GL}(n)\), a function that can be treated with the help of the Poisson summation formula, thus by classical methods. Their proposals were basically correct, but Tamagawa had confined himself, presumably for technical reasons, to the multiplicative groups of division algebras and Godement, perhaps also for technical reasons, perhaps because of a taste for functional analysis, wanted to introduce an auxiliary parameter.

I, myself, was searching for a general notion and had despaired. By the fall of 1966, I was prepared to abandon mathematics and to turn to some other life, a first step being a year or two in Turkey with my wife and children, as a prelude to an existence whose exact form was undetermined. I, who had never been anywhere outside of English-speaking North America, returned to the study of Russian and began the study of Turkish, frivolously daydreaming of a trip to Turkey --- with wife and four small children --- through the Balkans or through the Caucasus. In the end we arrived in Ankara by a more banal route. Even with the Russian and Turkish, I had time to spare and began, as an idle amusement, to calculate the constant term of the Eisenstein series for various rank-one groups. I had, curiously enough, never done this before. I discovered rather quickly a regularity of which I had been unaware. It was described in the lectures delivered at Yale some months later and included in Part 3 of this collection. The constant term, or rather the second part of the constant term, the part that expresses the functional equation was there denoted \(M(s)\) and given at the very end of §5 as a product that I write here as

\begin{equation} \prod_{i=1}^r \frac{\xi_i(a_is)}{\xi_i(a_is+1)} \tag{1} \end{equation} \(r\) being a small integer, often \(1\), and \(a_i\) being a positive number. Suppose, in order not to confuse the explanations, that \(r\) is \(1\). The issues arising in the general case are treated in the references. It is the relation expressed by (1) that suggests and allows the passage from the theory of Eisenstein series to a general notion of automorphic \(L\)-function that can accommodate not only a non-abelian generalization of class-field theory but also, as it turned out, both functoriality and reciprocity. It was the key to the suggestions in the Weil letter.

The Yale notes were written a long time ago and were hardly exemplary expositions. I have no desire at the moment to recall the details or to improve their presentation --- the reader is encouraged to consult the writings of Shahidi, for example the book Eisenstein series and automorphic \(L\)-functions --- but there are a number of points to which I would like to draw attention, and it is more convenient to refer to my own notes. I repeat, first of all, that (1) refers to rank-one parabolic subgroups, thus to Eisenstein series arising from maximal proper parabolic subgroups, so that it does not require the second or the third steps and is analytically at the level of Selberg's original arguments. Algebraically, however, one has to be thinking at the level of the theory of reductive groups. If I had not searched assiduously for a general form of the theorems of Hecke and of the founders of class field theory, or had not been familiar with various principles of nonabelian harmonic analysis as it had been developed by Harish-Chandra, in particular with the theory of spherical functions, I might have failed to recognize the importance or value of (1). At all events, between the discovery of (1) and the Yale lectures, I had no time for my linguistic undertakings, which were temporarily set aside, and by the summer of 1967, after the letter to Weil and the Yale lectures, I was again exhausted and was content to rest before departing with my family for Ankara.

Although the letter to Weil is not mentioned in those lectures, the critical discoveries that led quickly to the conjectures in that letter are described. They are all related to the formula (1). Before reviewing them, I recall that in the early sixties, thus before the letter to Weil and before the Yale lectures, a number of mathematicians had created a structure theory for groups over \(p\)-adic fields. These were described in the very successful Boulder conference organized by Borel and Mostow. Some of the representation theory for real groups, created by Harish-Chandra, by the Russian school and, to a lesser extent, by others, had been extended to groups over \(p\)-adic fields, in particular the theory of spherical functions to which among others, Satake had contributed. This was perhaps not very difficult, but it was certainly necessary. Satake describes the contribution of his paper, Theory of Spherical Functions on reductive algebraic groups over \(\frak p\)-adic fields, Publ. Math. IHES, 18 (1963), in the following words, ``Then our main theorem asserts that \(\mathcal L(G,U)\) is isomorphic to the algebra of all \(W\)-invariant polynomials functions on \(\mathrm{Hom}(M,\mathbf C^*)\simeq\mathbf C^\nu,\dots\); thus \(\mathcal L(G,U)\) is an affine algebra of (algebraic) dimension \(\nu\) over \(\mathbf C)\).'' The algebra \(\mathcal L(G,U)\) is the algebra of spherical functions. This is a clear, precise statement of an indispensable theorem, or lemma, a lemma that is, however, suggested immediately by the analogous lemma for spherical functions over the real field. The step from it to what I have called the Frobenius-Hecke conjugacy class rather than the Satake parameter --- a term often used by others --- in order to emphasize the importance of the contributions of these two outstanding mathematicians to the theory of automorphic \(L\)-functions, is technically minute, but entails a fundamental conceptual change that arises directly from the expression (1). To take the step it is only necessary to be aware that the space of coweights of a reductive group \(G\) is the space generated by the weights of a second group, again a reductive group and naturally associated to \(G\). Without understanding the formula (1), there is, however, no reason for taking it. Even serious mathematicians, for example, Benedict Gross --- see his exposition, On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry, London, Math. Soc. Lecture Notes, 254 --- fail to appreciate this. I tried in the two messages to Mueller and Volpato to explain how unexpected --- and perhaps undeserved --- it was that the Eisenstein series and their constant terms, once calculated, suggested the step and led to the introduction of the \(L\)-group and of the Frobenius-Hecke class into the theory of automorphic forms. At all events, the \(L\)-group and the Frobenius-Hecke conjugacy class do not appear in Satake's paper nor does the isomorphism of the Hecke algebra with the representation ring of \({}^LG\) as such, only with a ring to which it is isomorphic. There was no reason, in the context of the paper, that they should!

I have recalled the context in which they appeared. After having been introduced, in one way or another in the early 1960's to the papers of Siegel, Selberg, Hecke, and Harish-Chandra and to class-field theory and after having passed a free, but disappointing year 1964/65 in Berkeley, where I made an unsuccessful attempt to learn algebraic geometry and spent a good deal of time to not much purpose thinking about spherical functions in general and their relation with the classical hypergeometric functions, functions whose integral representations had intrigued me, I participated in the Boulder conference and learned, somewhat belatedly, to think in terms of reductive algebraic groups. So I had the background to reflect on the possibility not only of a non-abelian class field theory but also of attaching \(L\)-functions to automorphic forms or, better representations in general. My initial reflections were none the less, as already recalled, not successful.

Luckily I had not forgotten the problems and, when in the fall of 1966, as a pastime, I began to calculate the constant term of the Eisenstein series for rank-one groups, basically one group or one class of groups at a time, I noticed almost immediately expressions like formula (1) appearing. What is their value or significance? First of all, as they arise from Eisenstein series which are meromorphic in the whole \(s\)-plane, they themselves are meromorphic in the plane. Secondly, at least when \(r=1\), if the quotient \(\xi(as)/\xi(as+1)\) is meromorphic, then so is \(\xi(s)\). As observed, the cases in which \(r\neq 1\) can be treated by supplementary arguments. Moreover, \(\xi(s)\) is an Euler product and it is associated to the automorphic representation defining the Eisenstein series. Running through the pairs \((H,G)\) for which \(H\) was the Levi factor of a maximal proper parabolic subgroup of \(G\), I found that for all but three classes of simple \(H\) (at first I focussed on split groups) one could construct Euler products with meromorphic continuation. Clearly at this initial stage, after my earlier lack of success, meromorphic continuation was a triumph! Had I really arrived at the series for which I had been searching for so long? Was there a direct definition of these Euler products that depended only on the automorphic representation not on the construction of the Eisenstein series? I made a list of the groups \(H\), many of them classical groups, with familiar definitions and of the degree of the Euler products that arose from the several ways that \(H\) could appear as a Levi factor. The lists, published in the notes from the Yale lectures, revealed that the degrees were degrees not of representations \(\rho\) of \(H\), but of the group \({}^LH\) dual to it in the classification of semisimple groups or, more generally, reductive groups. The local factors were moreover immediately seen to be \(L(s,\pi,\rho)=1/\det(1-\rho(\gamma_p)/p^s)\), where \(\gamma\) was the parameter associated to the prime \(p\) and the unramified representation \(\pi_p\) of the local group \(H(\mathbf Q_p)\) (I was working over \(\mathbf Q\).) when one interprets the Satake isomorphism in terms of this duality. Here \(\rho\) was a representation of the group of \({}^LH\) that depended on the pair \((H,G)\). That this was not the way the isomorphism was interpreted by Satake is less important than the miracle of the appearance of these Euler products in the theory of Eisenstein series, although in a manner not immediately evident. Once their appearance was discovered, not only the resemblance to the Artin \(L\)-functions and the possibility of a non-abelian class field theory but also the generalizations of it envisaged in the Weil letter were evident. The miracle remains and it is this that I was trying to explain in the accompanying messages to Mueller/Volpato.

I add, as a supplementary remark, that I had calculated the factors of the Euler product for one pair \((H,G)\) at a time, and it was by no means a foregone conclusion that their degree would be given by the dimension of a representation \(\rho\) of \({}^LH\). Indeed, initially this was an empirical observation that led to the formal introduction of the group \({}^LH\). It was Tits who pointed out when I gave, during the lectures, the complete list of pairs \((H,G)\) and the associated \(\rho\) that a case-by-case verification was unnecessary, that \(\rho\) was the representation of \({}^LH\) on the unipotent radical of the dual pair \({}^L\mathfrak h \subset {}^L\mathfrak g\).

As a second comment, I also add that a question of convergence arose for the Euler products defining \(L(s,\pi,\rho)\) for an automorphic \(\pi\). To deal with it, it was necessary to give a formula for the spherical function associated to the local \(\pi_p(g)\), but the formula did not need to be everywhere valid. The formula with the necessary domain of validity was proven in my Yale notes. The same formula was later discovered independently and in a different context by Ian Macdonald, was proved by him for all \(g\), and is known as Macdonald's formula.

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