Armand Borel (1923–2003), Volume 51, Number 5

Armand Borel (1923–2003)

James Arthur, Enrico Bombieri, Komaravolu Chandrasekharan, Friedrich Hirzebruch, Gopal Prasad, Jean-Pierre Serre, Tonny A. Springer, and Jacques Tits

Citations within this article. Borel’s Collected Papers are [Œ], and his 17 books are referred to as [1] through [17]. These are all listed in a sidebar on page 501. An item like [Œ 23] is Borel paper number 23 in [Œ]. Ci- tations of work by people other than Borel are by letter combinations such as [Che], and the de- tails appear at the end of the article.

their root systems. He spent the year 1949–50 in 1

Institute for Advanced Study, 1999. Paris, with a grant from the CNRS . A good choice Armand Borel (for us, as well as for him), Paris being the very spot where what Americans have called “French Topol- Jean-Pierre Serre ogy” was being created, with the courses from The Swiss mathematician Armand Borel died Leray at the Collège de France and the Cartan sem- August 11, 2003, in Princeton from a rapidly evolv- inar at the École Normale Supérieure. Borel was an ing cancer. Few foreign mathematicians had as active participant in the Cartan seminar while many connections with France. He was a student closely following Leray’s courses. He managed to of Leray, he took part in the Cartan seminar, and understand the famous “spectral sequence”, not an he published more than twenty papers in collabo- easy task, and he explained it to me so well that I ration with our colleagues Lichnerowicz and Tits, have not stopped using it since. He began to apply as well as with me. He was a member of Bourbaki it to Lie groups, and to the determination of their for more than twenty years, and he became a for- cohomology with integer coefficients. That work would make a thesis, defended at the Sorbonne eign member of the Académie des Sciences in 1981. (with Leray as president) in 1952, and published im- French mathematicians feel that it is one of their mediately in the Annals of Mathematics. Meanwhile own who has died. Borel returned to Switzerland. He did not stay long. He was born in La Chaux-de-Fonds in 1923 and He went for two years (1952–54) to the Institute for was an undergraduate at Eidgenössische Technis- Advanced Study in Princeton and spent the year che Hochschule of Zürich (the “Poly”). There he met 1954–55 in Chicago, where he benefited from the H. Hopf, who gave him a taste for topology, and presence of André Weil by learning algebraic geom- E. Stiefel, who introduced him to Lie groups and etry and number theory. He returned to Switzer- Jean-Pierre Serre is professor emeritus of the Collège de land, this time to Zürich, and in 1957 the Institute France. His email address is [email protected]. This part of for Advanced Study offered him a position as per- the article is a translation of the presentation made by him manent professor, a post he occupied until his to the Académie des Sciences in Paris on September 30, death (he became a professor emeritus in 1993). 2003 (© Académie des Sciences and printed with permis- sion). Anthony W. Knapp assisted with the translation. 1Centre National de la Recherche Scientifique.

498 NOTICES OF THE AMS VOLUME 51, NUMBER 5 again, Borel’s extensive knowledge of the literature James Arthur played a role. Inspired by a recent paper of V. P. My topic is Armand Borel and the theory of au- Platonov, he had written a note proving the analog tomorphic forms. Borel’s most important contri- for simple groups over algebraically closed fields butions to the area are undoubtedly those estab- of a classical result of V. V. Morozov: a maximal lished in collaboration with Harish-Chandra [Œ 54, proper subalgebra of a semisimple complex Lie al- 58]. They include the construction and properties gebra either is semisimple or contains a maximal of approximate fundamental domains, the proof of nilpotent subalgebra. Borel sent a copy of the man- finite volume of arithmetic quotients, and the char- uscript of this note to a mutual friend of ours, acterization in terms of algebraic groups of those who showed it to me. I observed that the argument arithmetic subgroups that give compact quotients. of Borel’s proof could be adapted to a considerably These results created the opportunity for working more general setting and provided results over ar- in the context of general algebraic groups. They laid bitrary fields, for instance the following fact, which the foundations of the modern theory of automorphic forms that has flourished for the past turned out to be of great importance in finite group forty years. theory: if G is a reductive k-group (k being any field) 25 The classical theory of modular forms concerns and U is a split unipotent subgroup, then there holomorphic functions on the upper half plane H exists a parabolic k-subgroup P of G whose unipo- that transform in a certain way under the action tent radical contains U such that every k-auto- of a discrete subgroup Γ of SL(2, R). The multi- morphism of G preserving U also preserves P; plicative group SL(2, R) consists of the 2 × 2 real then, in particular, P contains the normalizer of U. matrices of determinant 1, and each element (Much earlier, I had conjectured this fact and given γ = ab of SL(2, R) acts on H by the linear cd a case-by-case proof “in most cases”, but the new az+b fractional transformation z → . For example, approach, using Borel’s argument, gave a uniform cz+d one can take Γ to be the subgroup SL(2, Z) of and much simpler proof.) integral matrices or, more generally, the congruence I mention en passant the complements [Œ 94] subgroup to “Groupes réductifs”, containing results con- ∈ Z ≡ cerning, among other things, the closure of Bruhat Γ (N)= γ SL(2, ): γ I mod N cells in topological reductive groups and the fun- attached to a positive integer N. The theory began damental group of real algebraic simple groups. as a branch of complex analysis. However, with the The last joint paper by Borel and me is a Comptes work of E. Hecke, it acquired a distinctive number Rendus note [Œ 97] that in personal discussions we theoretic character. Hecke introduced a commut- nicknamed “Nonreductive groups”. Indeed, we had ing family of linear operators on any space of au- discovered that many of our results on reductive tomorphic forms for Γ (N), one for each prime not groups could, after suitable reformulation, be gen- dividing N, with interesting arithmetic properties. eralized to arbitrary connected algebraic groups. We now know that eigenvalues of the Hecke operators govern how prime numbers p split in certain Examples are the conjugacy by elements rational nonabelian Galois extensions of the field Q of ra- over the ground field, of maximal split tori, of tional numbers [Sh], [D]. Results of this nature are maximal split unipotent subgroups, and of mini- known as reciprocity laws and are in some sense 26 mal pseudoparabolic subgroups, or the existence the ultimate goal of algebraic number theory. They of a BN-pair (hence of a Bruhat decomposition). can be interpreted as a classification for the num- Complete proofs of those results never got pub- ber fields in question. The Langlands program con- lished, but both Borel and I, independently, lectured cerns the generalization of the theory of modular about them (he at Yale University, and I at the Col- forms from the group of 2 × 2 matrices of deter- lège de France). minant 1 to an arbitrary reductive group G. It is believed to provide reciprocity laws for all finite algebraic extensions of Q. Let us use the results of Borel and Harish- Chandra as a pretext for making a very brief ex- cursion into the general theory of automorphic forms. In so doing, we can follow a path illuminated 25 A unipotent group over a field k is said to be split if it by Borel himself. The expository articles and mono- k has a composition series over , all quotients of which are graphs of Borel encouraged a whole generation of additive groups. 26 Pseudoparabolic subgroups are a substitute for para- James Arthur is professor of mathematics at the Univer- bolic subgroups that one must use when dealing with sity of Toronto and president elect of the AMS. His email nonreductive groups over nonperfect fields. address is [email protected].

518 NOTICES OF THE AMS VOLUME 51, NUMBER 5 mathematicians to pursue the study of automor- organized jointly by Borel and G. D. Mostow. It phic forms for general algebraic groups. Together was a systematic attempt to make the emerging the- with the mathematical conferences he organized, ory of automorphic forms accessible to a wider au- they have had extraordinary influence. dience. Borel himself wrote four articles [Œ 73, 74, The general theory entails two reformulations 75, 76] for the proceedings, each elucidating a dif- of the classical theory of modular forms. The first ferent aspect of the theory. is in terms of the unitary representation theory of The second reformulation is in terms of adeles. the group SL(2, R). Though harder to justify at first, the language of The action of SL(2, R) on H√ is transitive. Since adeles ultimately streamlines many fundamental the stabilizer of the point i = −1 is the special or- operations on automorphic forms. The relevant thogonal group Boulder articles are [T] and [K]. These were not cos θ sin θ written by Borel, but were undoubtedly commis- KR = SO(2, R)= k = , θ − sin θ cos θ sioned by him as part of a vision for presenting a one can identify H with the space of cosets coherent background from the theory of algebraic SL(2, R)/KR. The space of orbits in H under a dis- groups. crete group Γ ⊂ SL(2, R) becomes the space of dou- The adele ring ∼ ble cosets Γ \SL(2, R)/KR. A modular form of weight A = R × A = R × Q 2k is a holomorphic function f on H such that27 ﬁn p p prime 2k f (γz)=(cz + d) f (z) of Q is a locally compact ring that contains the real field R, as well as completions Qp of Q with respect whenever γ = ab is in Γ. A modular form of cd to the p-adic absolute values weight 2, for instance, amounts to a holomorphic −r r −1 1-form f (z)dz on the Riemann surface Γ \H, since |x|p = p ,x= p (ab ), (a, p)=(b, p)=1, d(γz)=(cz + d)−2dz. For a given f, the function F on Q. One constructs Q by a process identical to on SL (2,R) defined by F(g)=(ci + d)2kf (z) when p the completion R of Q with respect to the usual ab- g = ab and z = gi is easily seen to satisfy cd solute value. In fact, one has an enhanced form of −2kiθ the triangle inequality, F(γgkθ)=F(g)e , for γ ∈ Γ. The requirement that f be holomorphic |x1 + x2|p ≤ max |x1|p, |x2|p ,x1,x2 ∈ Q, translates to the condition that F be an eigen- which has the effect of giving the compact “unit function of a canonical biinvariant differential op- ball” Z = x ∈ Q : |x | ≤ 1 the structure of erator ∆ on SL(2, R) of degree 2, with eigenvalue p p p p p a simple function of k. The theory of modular a subring of Qp. The complementary factor Aﬁn forms of any weight becomes part of the follow- of R in A is defined as the “restricted” direct ing more general problem: product ∼ Decompose the unitary representation Qp = x =(xp): xp ∈ Qp, of SL(2, R) by right translation on p L2 Γ \SL(2, R) into irreducible repre- xp ∈ Zp for almost all p , sentations. That the problem is in fact more general is due which becomes a locally compact (totally discon- to a variant of Schur’s lemma. Namely, as an op- nected) ring under the appropriate direct limit erator that commutes with SL(2, R), ∆ acts as a topology. The diagonal image of Q in A is a discrete scalar on the space of any irreducible representa- subring. This implies that the group SL(2, Q) of ration. To recover the modular forms of weight 2k, tional matrices embeds into the locally compact A one would collect the irreducible subspaces of group SL(2, ) of unimodular adelic matrices as a L2 Γ \SL(2, R) with the appropriate ∆-eigenvalue, discrete subgroup. The theory of automorphic forms on Γ \SL(2, R), for any congruence subgroup and from each of these, extract the smaller subspace Γ = Γ (N), becomes part of the following more gen- on which the restriction to KR of the corresponding eral problem: SL(2, R)-representation equals the character −2kiθ Decompose the unitary representation kθ → e . This is all explained clearly in Borel’s survey ar- of SL(2, A) by right translation on ticle [Œ 75] in the proceedings of the 1965 AMS con- L2 SL(2, Q)\SL(2, A) into irreducible ference at Boulder [3]. The Boulder conference was representations. The reason that the last problem is more gen- 27There is also a mild growth condition that need not eral than the previous one is provided by the the- concern us here. orem of strong approximation, which applies to the

MAY 2004 NOTICES OF THE AMS 519 Figure 1. Standard fundamental domain for the Photograph by Lyn Howe, provided courtesy of Roger Howe. action of SL(2, Z) on the upper half plane, “Armand Borel Math Camp” at the AMS Summer together with a more tractable approximate Institute on automorphic forms, representations, and fundamental domain. The standard L-functions at Corvallis, Oregon, 1977. Front: Nick fundamental domain, darkly shaded, is the Howe. Back row, left to right: Roger Howe, Armand semi-infinite region bounded by the unit circle Borel, Robert Langlands, Bill Casselman, Marie-France and the vertical lines at x =±1/2. The Vigneras, Kenneth Ribet. Borel has the special approximate fundamental domain St designation “Coach” on his T-shirt and Casselman has generalized by Borel and Harish-Chandra is the the designation “Assistant Coach”. total shaded region. 28 simply connected group SL(2). The theorem as- SL(2, Aﬁn). The unitary representation theory of the serts that if K is any open compact subgroup of p-adic groups SL(2, Qp) thus plays an essential SL(2, Aﬁn), then role in the theory of modular forms. This is the source of the operators discovered by Hecke. SL(2, A)=SL(2, Q) · (K · SL(2, R)). Eigenvalues of Hecke operators are easily seen to This implies that if Γ = SL(2, R) ∩ SL(2, Q)K, then parametrize irreducible representations of the Q there is a unitary isomorphism group SL(2, p) that are unramified in the sense that their restrictions to the maximal compact ∼ L2 SL(2, Q)\SL(2, A)/K −→ L2 Γ \SL(2, R) subgroup SL(2, Zp) contain the trivial representation. It turns out that in fact any irreducible that commutes with right translation by SL(2, R). representation of SL(2, Qp) that occurs in the For example, if we take K to be the group decomposition of L2 SL(2, Q)\SL(2, A) carries fundamental arithmetic information. K(N)= x =(xp): xp ∈ SL(2, Zp), It is now straightforward to set up higher- xp ≡ I mod (M2(NZp)) , dimensional analogs of the theory of modular forms. One replaces29 the group SL(2) by an arbi- then Γequals Γ (N). To recover the decomposition trary connected reductive algebraic group G de- 2 of L Γ (N)\SL(2, R) , one would collect the fined over Q. As in the special case of SL(2), G(Q) irreducible subspaces of L2 SL(2, Q)\SL(2, A) , embeds as a discrete subgroup of the locally com- A and from each of these, extract the smaller sub- pact group G( ). The Langlands program has to do space for which the restriction to K of the corre- with the irreducible constituents (known as automorphic representations) of the unitary represen- sponding SL(2, A)-representation is trivial. tation of G(A) by right translation on Given that the decomposition of L2 G(Q)\G(A) . A series of conjectures of Lang- L2 SL(2, Q)\SL(2, A) includes the classical lands, dating from the mid-1960s through the theory of modular forms, we can see reasons why 1970s, characterizes the internal structure of au- the adelic formulation is preferable. It treats the tomorphic representations. The conjectures provide theory simultaneously for all weights and all con- a precise description of the arithmetic data in au- gruence subgroups. It is based on a discrete group tomorphic representations, together with a for- SL(2, Q) of rational matrices rather than a group mulation of deep and unexpected relationships Γ (N) of integral matrices. Most significantly, per- among these data as G varies (known as the “prin- haps, it clearly displays the supplementary struc- ciple of functoriality”). ture given by right translation under the group 29Even in the classical case, one has to replace SL(2) by 28“Simply connected” in this instance means that SL(2, C) the slightly larger group GL(2) to obtain all the opera- is simply connected as a topological space. tors defined by Hecke.

520 NOTICES OF THE AMS VOLUME 51, NUMBER 5 General automorphic representations are thus only finitely many γ ∈ SL(2, Z) such that St and γSt firmly grounded in the theory of algebraic groups. intersect. It seems safe to say that the many contributions For a general group G, the results of Borel and of Borel to algebraic groups described by Springer Harish-Chandra provide an approximate funda- and Tits in this article are all likely to have some mental domain for the action of G(Q) by left trans- role to play in the theory of automorphic forms. lation on G(A). To describe it, I have to rely on a Borel did much to make the Langlands program few notions from the theory of algebraic groups. more accessible. For example, his Bourbaki talk Let me write P for a minimal parabolic subgroup [Œ 103] in 1976 was one of the first comprehen- of G over Q, with unipotent radical N and Levi com- sive lectures on the Langlands conjectures to a ponent M. The adelic group M(A) can be written 1 0 general mathematical audience. as a direct product M(A) AM (R) , where AM is the 0 In 1977 Borel and W. Casselman organized the Q-split part of the center of M, AM (R) is the con- 1 AMS conference in Corvallis on automorphic forms nected component of 1 in AM (R), and M(A) is a 0 and L-functions, as a successor to the Boulder con- canonical complement of AM (R) in M(A) that conference. It was a meticulously planned effort to pre- tains M(Q). The roots of (P,AM ) are characters α sent the increasingly formidable background ma- a−→ a on AM that determine a cone terial needed for the Langlands program. The ∈ R 0 α ≥ Corvallis proceedings [10] are considerably more At = a AM ( ) : a t, for every α challenging than those of Boulder. However, they 0 in A (R) for any t>0. Suppose that KA = K K remain the best comprehensive introduction to M R ﬁn is a maximal compact subgroup of G(A). If Ω is a the field. They also show evidence of Borel’s firm compact subset of N(A)M(A)1, the product hand. Speakers were not left to their own devices. On the contrary, they were given specific advice on St = ΩAt KA exactly what aspect of the subject they were being A asked to present. Conference participants actually is called a Siegel set in G( ), following special cases had to share facilities with a somewhat unsympa- introduced by C. L. Siegel. One of the principal re- thetic football camp, led by Coach Craig Fertig of sults of [Œ 58] implies that for suitable choices of the Oregon State University Beavers. At the end of KA , Ω, and t, the set St is an approximate funda- Q A the four weeks, survivors were rewarded with or- mental domain for G( ) in G( ). Q \ A ange T-shirts, bearing the inscription ARMAND The obstruction to G( ) G( ) being compact is BOREL MATH CAMP. Borel himself sported30 a thus governed by the cone At in the group R 0 ∼ Rdim AM Q \ A T-shirt with the further designation COACH. AM ( ) = . It follows that G( ) G( ) is Let us go back to the topic we left off earlier, compact if and only if AM is trivial, which is to say Q Borel’s work with Harish-Chandra. The action of that G has no proper parabolic subgroup over Z H and no Q-split central subgroup. This is essen- SL(2, ) on has a well-known fundamental do- 31 main, given by the darker shaded region in tially the criterion of [Œ 58]. Borel and Harish- Chandra obtained other important results from Figure 1. This region is difficult to characterize in their characterization of approximate fundamen- terms of the transitive action of SL(2, R) on H. The tal domains. For example, in the case of semisim- total shaded rectangular region S in the diagram t ple G, they proved that the quotient G(Q)\G(A) is more tractable, for there is a topological de- has finite volume with respect to the Haar measure composition SL(2, R)=P(R)KR = N(R)M(R)KR , of dx on G(A). This is a consequence of a decom- where P, N, and M are the subgroups of matrices position formula for Haar measures in SL(2) that are respectively upper triangular, dx = a−2ρ dωdadk, where ω is in Ω, a is in A , upper triangular unipotent, and diagonal. The t and k is in KA and where 2ρ denotes the sum of group N(R) acts by horizontal translation on H, the roots of (P,A ). while M(R) acts by vertical dilation. We have already M The papers of Borel and Harish-Chandra were noted that KR stabilizes the point i. We can there- actually written for arithmetic quotients Γ \G(R) fore write of real groups, as were the supplementary articles St = ωAt · i, [Œ 59, 61] of Borel. Prodded by A. Weil [W, p. 25], where ω is the compact subset 1 x : |x|≤σ 01 Borel wrote two parallel papers [Œ 55, 60] that of N(R), and At is the one-dimensional cone formulated the results in adelic terms and estab- 32 a 0 : a>0,a2 ≥ t . The set S is an lished many basic properties of adele groups. His 0 a−1 t approximate fundamental domain for the action of 31A similar result was established independently by Mostow SL(2, Z) on H, in the sense that it contains a set and Tamagawa [MT]. of representatives of the orbits, while there are 32In his 1963 Bourbaki lecture [G], R. Godement presented an alternative argument, which he also formulated in 30See the photograph on the previous page. adelic terms.

MAY 2004 NOTICES OF THE AMS 521 later lecture notes [Œ 79], written again in the between intersection cohomology (discovered by setting of real groups, immediately became a stan- Goresky and MacPherson in the 1970s) and L2- dard reference. cohomology (applied to square integrable Borel and Harish-Chandra were probably moti- differential forms on XΓ), a relationship first con- 33 2 vated by the 1956 paper [Sel] of A. Selberg. Selberg jectured by S. Zucker. The L -cohomology of XΓ brought many new ideas to the study of the spec- is the appropriate analog of deRham cohomology 2 tral decomposition of L Γ \SL(2, R) , including a in case XΓ is noncompact. Its relations with auto- construction of the continuous spectrum by means morphic forms were investigated by Borel and Cas- of Eisenstein se- selman [Œ 126, 131]. In general, the cohomology ries and a trace groups of spaces XΓ are very interesting objects, formula for ana- which retain many of the deepest properties of lyzing the dis- the corresponding automorphic representations. crete spectrum. They bear witness to the continuing vitality of A familiarity with mathematics that originated with Borel. the results of Siegel no doubt gave Borel and Harish-Chandra Gopal Prasad encouragement Borel first visited India in 1961, when he gave a for working with series of lectures on compact and noncompact

Photograph by C. J. Mozzochi. general groups. semisimple Lie groups and symmetric spaces at the Their papers Werner Mueller, Leslie Saper, and Borel Tata Institute of Fundamental Research (TIFR) in were followed in (left to right) at the Park City Summer Bombay. His course introduced the theory of Lie the mid-1960s Institute, 2002. groups and geometry of symmetric spaces to the by Langlands’s first generation of mathematicians at TIFR, who in manuscript on turn trained the next generations in these areas. general Eisenstein series (published only later in Having joined TIFR in 1966, I belong to the second 1976 [L]). In the context of adele groups, Lang- generation, and so I owe a considerable debt to Ar- lands’s results give a complete description of the mand Borel for my education in the theory of Lie 2 Q \ A continuous spectrum of L G( ) G( ) . A starting groups and even more directly in the theory of al- point was the work of Borel and Harish-Chandra gebraic and arithmetic groups, which I learned and, in particular, the properties of approximate from his excellent books and numerous papers on fundamental domains. In recent years Borel lectured these two topics. widely on the theory of Eisenstein series: in the Subsequently, Borel made many visits to TIFR three-year Hong Kong program mentioned by and other mathematical centers in India. He was Bombieri, for example, and the 2002 summer school deeply interested in the development of mathe- in Park City. One of his ambitions, alas unrealized, matical institutions in India and helped several of was to write an introductory volume on the gen- them with his advice and frequent visits. For his eral theory of Eisenstein series. contributions to Indian mathematics, he was made In attempting to give a sense of both the scope an honorary fellow of TIFR in 1990. of the field and Borel’s substantial influence, I have During his numerous trips to India, Armand emphasized Borel’s foundational work with Harish- and his wife, Gaby, visited many sites of historical Chandra and his leading role in making the sub- and architectural interest and attended concerts of ject more accessible. Borel made many other im- Indian classical dance and music. He became fond portant contributions. These were often at the of many forms of Indian classical dances such as interface of automorphic forms with geometry, es- Bharat-Natyam, Odissi, and Kathakkali. Even more pecially as it pertains to the locally symmetric so, he came to love both Carnatic (south Indian) and spaces Hindustani (north Indian) classical music. He be- X = Γ \G(R)/KR . Γ came quite an expert on the subject and devel- Elements in the deRham cohomology group oped friendships with many musicians and dancers ∗ H (XΓ , C) are closely related to automorphic forms from India. He invited several of them to perform for G, as we have already noted in the special case in the concert series he initiated and directed at the of modular forms of weight 2. This topic was fully explored in Borel’s monograph [Œ 115, 172] with N. Wallach. Borel collaborated in the creation of two 33The conjecture was later proved by L. Saper and M. Stern, very distinct compactifications of spaces XΓ: one and E. Looijenga. with W. Baily [Œ 63, 69], the other with J-P. Serre Gopal Prasad is professor of mathematics at the Univer- [Œ 90, 98]. The Baily-Borel compactification be- sity of Michigan at Ann Arbor. His email address is came the setting for the famous correspondence [email protected].

522 NOTICES OF THE AMS VOLUME 51, NUMBER 5