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Group Representations and Harmonic Analysis from Euler to Langlands, Part I Anthony W Group Representations and Harmonic Analysis from Euler to Langlands, Part I Anthony W. Knapp roup representations and harmonic for s>1. In fact, if each factor (1 p s ) 1 on − − − analysis play a critical role in subjects the right side of (1) is expanded in geometric se- as diverse as number theory, proba- ries as 1+p s +p 2s + , then the product of − − ··· bility, and mathematical physics. A the factors for p N is the sum of those terms ≤ representation-theoretic theorem of 1/ns for which n is divisible only by primes GLanglands is a vital ingredient in the work of N; hence a passage to the limit yields (1). Wiles on Fermat's Last Theorem, and represen- ≤ Euler well knew that the sum for (s) exceeds tation theory provided the framework for the the integral prediction that quarks exist. What are group representations, why are they so pervasive in ∞ dx 1 s = mathematics, and where is their theory headed? 1 x s 1. Z − Euler and His Product Expansions This expression is unbounded as s decreases to Like much of modern mathematics, the field of 1, but the product (1) cannot be unbounded un- group representations and harmonic analysis less there are infinitely many factors. Hence (1) has some of its roots in the work of Euler. In 1737 yielded for Euler a new proof of Euclid's theo- Euler made what Weil [4] calls a “momentous dis- rem that there are infinitely many primes. In covery”, namely, to start with the function that fact, (1) implies, as Euler observed, the better the- we now know as the Riemann function 1 orem that 1/p diverges. (s)= ∞ s and to realize that the sum could n=1 n Euler later went on from this proof to deduce be written as a product P P that there are infinitely many primes 4n +1and 1 (1) (s)= infinitely many primes 4n +3, and that is where 1 p s pprimeY − − the story of harmonic analysis really begins. To understand why the above analysis does not handle these cases, it is helpful to see in more detail how 1/p enters the above kind of ar- Anthony W. Knapp is professor of mathematics at the gument. Consideration of series expansions of State University of New York, Stony Brook. His e-mail P address is [email protected]. the exponentiated functions shows that 1 log(1 + x) <x<log 1 x The author expresses his appreciation to Sigurdur Hel- − gason, Hugo Rossi, John Tate, David Vogan, and the In- for 0 <x<1, and it is easy to see that the right stitut Mittag-Leffler for help in the preparation of this article. Part II of this article will appear in the May 1996 side is no more than twice the left side if 0 <x 1. Consequently it follows from (1) that issue of the Notices. ≤ 2 410 NOTICES OF THE AMS VOLUME 43, NUMBER 4 1 1 1 1 ∞ χ−(n) (2) log (s) = s + bounded term L(s) = 1 + + = p − 3s 5s − 7s · · · ns p primeX n=1 (7) X as s decreases to 1. Meanwhile, multiplication 1 = s , of the series for (s) by 2 s reproduces the 1 χ−(p)p− − p primeY − even-numbered terms of the series, and there- fore where χ−(n) is 0 for n even, 1 for n 1 mod 4, ≡ 1 1 1 1 and 1 for n 3 mod 4. The log of s is (3a) 1 (s) = 1 + + + 1 χ(p)p− s s s − ≡ s − − 2 3 5 · · · approximately χ(p)p− even if χ(p) is negative. and 2 Arguing by taking the log of the product formula 1 (s) = − 2s in (6) (or simply copying the result from (4)) (3b) 1 1 1 1 1 + + . gives − 2s 3s − 4s 5s − · · · sum of terms in (5) = (8) 1 The left side of (3b) is (s 1)(s) times some- log + bounded term − s 1 thing that tends to log 2 as s tends to 1. Euler − knew Leibniz's test for convergence and could as s decreases to 1. Meanwhile application of the see in (3b) that the series on the right is con- Leibniz test to the series in (7) proves that L(s) vergent for s > 0 with a positive sum. It follows converges for s > 0 and in particular is finite at that (s) near s = 1 is the product of (s 1) 1 − − s = 1. In addition, the test shows that L(1) > 0. and a function with a finite nonzero limit. Com- Consequently taking the log of the product for- bining this result with (2) gives mula in (7) yields 1 1 (4) = log + bounded term ps s 1 (9) difference of terms in (5) = bounded term p primeX − as s decreases to 1. as s decreases to 1. Comparing (8) and (9) shows In handling primes congruent to 1 or 3 mod- that each of the series in (5) is unbounded as s ulo 4, it is tempting to replace the sum over all decreases to 1. Hence there are infinitely many primes of 1/ps in the above argument by primes congruent to 1 modulo 4 and also infi- nitely many primes congruent to 3. 1 1 (5) or , ps ps Role of a Group in Euler's Products p 1Xmod4 p 3Xmod4 ≡ ≡ Where is the group and what is its role? The + trace backwards, and see what happens. What property of the two functions χ and χ−, call ei- happens is that the expansion of the corre- ther of them χ, that allows the sums in (6) and sponding product of (1 p s ) 1 as a sum does (7) to be rewritten as products is that − − − not yield anything very manageable. Euler's key χ(mn) = χ(m)χ(n) for all positive integers m and new idea was to work with the sum and differ- n. Nowadays such functions are called Dirichlet + ence of the two terms in (5), rather than the two characters modulo 4. We can think of χ and χ− terms separately, and then to recover the two as lifts to the integers of functions on the mul- terms (5) at the end. This is full-fledged har- tiplicative group 1, 3 of integers modulo 4 and { } monic analysis on a 2-element group. prime to 4, with 0 used as the value on integers The essence of harmonic analysis is to de- that are not prime to 4. The two functions on the compose complicated expressions into pieces group 1, 3 are { } that reflect the structure of a group action when there is one; the goal is to make some difficult ω+(1) = ω+(3) = +1 and analysis manageable. Tracing backwards with ω−(1) = +1, ω−(3) = 1. the earlier argument as a model, Euler found two − manageable series with product expansions. The These functions ω on this 2-element group are first was multiplicative characters, i.e., homomorphisms 1 1 1 1 to the multiplicative group of nonzero complex 1 s (s) = 1 + s + s + s + numbers, and they are the only multiplicative − 2 3 5 7 · · · (6) characters for this group. They form a basis for ∞ χ+(n) 1 = = , the complex vector space of all complex-valued ns 1 χ+(p)p s nX=1 p primeY − − functions on the 2-element group. Essentially Euler had two functions to study, the char- where χ+(n) is 0 for n even and 1 for n odd. The acteristic function of each 1-element set for second series was this group: APRIL 1996 NOTICES OF THE AMS 411 through it and proved his own theorem about Euler primes in arithmetic progressions. The above work of Euler is of more than his- torical interest. It is the direct ancestor of a large amount of current research in algebraic number theory, including the representation- theoretic input from the Langlands program into Fermat's Last Theorem. In addition, it il- lustrates the principle that although harmonic analysis may be at the core of the solution of a problem, several layers of ingenious ideas may lie between the statement of the problem and the use of harmonic analysis. Multiplicative characters played a rather in- cidental role in mathematics from 1737 until about 1807. Cramer introduced determinants in 1750, defining the sign of a permutation and proving what we now call Cramer's Rule. The sign of a permutation is a multiplicative character on the permutation group on n letters, and deter- minant is a multiplicative character on nonsin- gular matrices of a fixed size. But the harmonic analysis aspect of these characters played no role in Cramer's work. Gauss, in expanding on Euler's work representing integers by binary quadratic Photograph courtesy of the Institut Mittag-Leffler forms, introduced his own notion of character, Leonhard Euler which corresponds roughly to what we now call a Dirichlet character. But again harmonic analy- I (1) = 1, I (3) = 0 and I (1) = 0, I (3) = 1. 1 1 3 3 sis was not involved. The series under study in (5) may be written as Fourier Series I (p) I (p) The next big development in the subject of group 1 and 3 . ps ps representations was the subject of Fourier series. p primeX p primeX The account here is taken from Grattan-Guinness [1]. In 1747 d'Alembert presented his work on Euler's proof worked because he expanded the the vibrating string problem: He found the dif- functions I1 and I3 in terms of the basis of mul- ferential equation tiplicative characters ∂2y 1 ∂2y = , 1 + 1 + ∂x2 c2 ∂t2 I1 = (ω + ω−) and I3 = (ω ω−), 2 2 − succeeded at some computations for the indi- specified initial conditions, and obtained the solution y = 1(f(x + ct) + f(x ct)) .
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