sign in sign up
<<
Home , ATLAS of Finite Groups, Character (mathematics), Character table, Character theory, David Vogan, E7 (mathematics)

The Table for

n January 8, 2007, just before 9 in What’s E8? the morning, a computer finished writ- A Lie is a group endowed with the structure ing to disk about sixty gigabytes of of a smooth manifold, in such a way that group files containing the Kazhdan-Lusztig multiplication and inversion are smooth maps. Opolynomials for the split real group Every finite group is a : the manifold G of type E8. Values at 1 of these polynomials structure is just the zero-dimensional discrete are coefficients in characters of irreducible repre- structure. For that reason the study of Lie groups sentations of G; so all irreducible characters were is necessarily more complicated than the study written down. The biggest coefficient appearing of finite groups. But it’s not unreasonable to was 11,808,808, in the polynomial concentrate on connected Lie groups. If you do that, a miracle happens: connected Lie groups are 152q22 + 3472q21 + 38791q20 + 293021q19 less complicated than finite groups. The reason is that a connected Lie group is almost completely +1370892q18 + 4067059q17 + 7964012q16 determined by its . The Lie algebra is the tangent space to the group manifold at the identity 15 14 13 +11159003q + 11808808q + 9859915q element, endowed with a nonassociative product called the Lie bracket. Since the Lie algebra is a +6778956q12 + 3964369q11 + 2015441q10 finite-dimensional , it can be studied using linear algebra ideas. A typical method is to +906567q9 + 363611q8 + 129820q7 look at one element X of the Lie algebra, and to +41239q6 + 11426q5 + 2677q4 regard “Lie bracket with X” as a linear transfor- mation from the Lie algebra to itself. This linear +492q3 + 61q2 + 3q. transformation has eigenvalues and eigenvectors, and those invariants can be used to describe the Its value at 1 is 60,779,787. structure of the Lie algebra. This calculation is part of a larger project Just as finite groups are successive extensions called the atlas of Lie groups and repre- of nonabelian simple groups (and Z/pZ), connected sentations. In this article I’ll try to explain the Lie groups are successive extensions of connected atlas project, what Kazhdan-Lusztig polynomials simple Lie groups (and the additive group R). Many are, why one might care about them, and some- questions about general Lie groups can be reduced thing about the nature of this calculation and the to the case of connected simple Lie groups, and calculators. so to questions about simple Lie algebras over the real numbers. David Vogan is professor of mathematics at the Mas- Many algebra problems are easier over alge- sachusetts Institute of Technology. His email address is braically closed fields, so it’s natural to relate Lie [email protected]. Supported in part by NSF FRG grant algebras over R to Lie algebras over C. If k ⊂ K is 0554278. any field extension, an n-dimensional Lie algebra

1022 Notices of the AMS Volume 54, Number 9 gk over the small field k gives rise naturally to an well established as the “right” way to present the n-dimensional Lie algebra gK = gk ⊗k K over the theory, the older explicit constructions continue large field K. The algebra gk is called a k-form of to be a powerful tool. gK . We can study real Lie algebras by first studying What we are seeking is an understanding of complex Lie algebras, and then studying their real Lie groups and (infinite-dimensional) representa- forms. tions of this aesthetically inferior sort: one that in 1887 was able to classify for the exceptional groups in particular may rely simple Lie algebras over the complex numbers. He on explicit calculations. Always our goal is to found four infinite families of classical Lie algebras find formulations of results that are as clean and An, Bn, Cn, and Dn; and five exceptional Lie algebras simple and general as possible; but we allow for G2, F4, , , and E8. Each of these complex Lie the possibility of verifying some results by long algebras has a finite number of real forms; the real computations. The calculation we have done for E8 forms were described completely by Élie Cartan in is certainly long. I will say a few words in the last the 1890s. section about the kind of clean and simple results In each case there are two distinguished real we are extracting from it. forms: the compact , for which the cor- A general warning about mathematical . responding Lie group is compact, and the split I have tried to make the mathematical statements real form. The term “split” refers to factorization convey accurately our level of understanding; but I of certain characteristic polynomials. The split have deliberately omitted or obscured many impor- form has the property that there is an open set tant details. (Here is an example. In equation (D) of Lie algebra elements X for which the linear below, I say that Harish-Chandra found a for transformation of Lie bracket with X has only real the solutions of a system of differential equations. eigenvalues. If one works with simple Lie algebras When certain eigenvalues for the system are zero, over other fields, there is always an analogue of Harish-Chandra did not find all the solutions. The the split form. The compact form is special to the ones that he found suffice for the expression real field. of irreducible characters: equation (E) remains The Lie groups attached to classical Lie algebras true.) Undoubtedly the number of unintentional are all related to classical linear algebra and ge- obscurities and omissions is equally large. For both ometry. A real Lie group of type Bn, for instance, categories, I apologize in advance. 2n+1 is the group of linear transformations of R The level of historical precision is perhaps even preserving a nondegenerate quadratic form. These lower. I have omitted reference to many mathe- groups were already known in Lie’s work, and in maticians whose work played a crucial role in the some sense they go back even to Euclid. developments reported here. For these omissions I The great surprise in Killing’s work was his will not attempt to give an illustrative example but discovery of the exceptional Lie algebras: five I will again apologize. simple Lie algebras (of 14, 52, 78, 133, and 248) having no straightforward connection to Unitary Representations and Their classical . Work in the twentieth century on the classification of finite simple groups shows Disreputable Cousins that we should be delighted with such a short and A unitary representation of a tractable list of exceptions. G is a continuous action of G by automorphisms The atlas project is aimed at understanding the of a Hilbert space (preserving the inner product). structure and of Lie groups. Another way to say this is that a unitary repre- Each member of the project has a slightly different sentation is a realization of G as symmetries of a idea about what “understanding” means. Certainly (possibly infinite-dimensional) Euclidean geometry. it should include a thorough understanding of the Because Hilbert spaces are the basic objects of exceptional groups. quantum mechanics, one can also say that a unitary There is an aesthetic in this subject according representation is a realization of G as symmetries to which the best proof is one not referring to of a quantum-mechanical system. A unitary repre- the Cartan-Killing classification. In practice, such a sentation is called irreducible if the Hilbert space proof may be difficult to find. One of the most fun- has exactly two closed G-stable subspaces. damental results is Cartan and Weyl’s description Because so many spaces are closely of the finite-dimensional irreducible representa- related to Hilbert spaces, unitary representations tions of a connected Lie group. This theorem are a fundamental tool for understanding ac- was first proved in the 1930s using explicit con- tions of topological groups. To prepare this tool structions of representations of simple Lie groups, for use, we seek to understand arbitrary unitary given separately in each case of the classification. representations of arbitrary topological groups. Only twenty years later did Harish-Chandra give An arbitrary unitary representation can often a construction independent of the classification. be written as a direct integral of irreducible uni- Even today, when Harish-Chandra’s approach is tary representations. The notion of direct integral

October 2007 Notices of the AMS 1023 extends that of . If T is the unit circle, (The correct notion of equivalence—Harish- the Hilbert space L2(T) has a direct sum decom- Chandra’sinfinitesimal equivalence—isabitsubtle, 2 P inθ position L (T) = n∈Z C · e given by Fourier and involves unbounded operators. That causes series. The Hilbert space L2(R) has a direct integral difficulties with making precise statements in the 2 R ixξ decomposition L (R) = ξ∈R C · e dξ given by the rest of this section, but nothing insurmountable.) . Quasisimple representations turn out to be easier There is an extremely general theorem guaran- to describe than unitary representations. teeing existence of a direct integral decomposition What is the relation to the original problem into irreducible representations: it suffices that of describing unitary representations? A unitary the topological group have a countable dense representation is automatically quasisimple, so we subset. There are moderately general theorems want to understand when a quasisimple irreducible guaranteeing uniqueness of direct integral decom- representation is actually unitary. That is, we want positions. This uniqueness holds (for example) to know whether Vπ admits a G-invariant Hilbert for all algebraic Lie groups— of n × n space structure. This question can be broken into matrices defined by polynomial equations in the two parts: whether there is a G-invariant Hermitian entries. We therefore seek to understand form, and whether this form is definite. We can irreducible unitary representations for algebraic therefore define Lie groups. Πh(G) = equivalence classes of irreducible Work begun by George Mackey and complet- quasisimple Hermitian representations of G, ed by Michel Duflo describes irreducible unitary representations of algebraic Lie groups by a very and get inclusions concrete and explicit reduction to the case of Πu(G) ⊂ Πh(G) ⊂ Πq(G). reductive algebraic groups. (These are essentially direct products of simple and abelian Lie groups.) The atlas goal of understanding the irreducible One of the goals of the atlas project is this: unitary representations of a reductive algebraic Lie group G can now be divided into three steps: to describe the set (G) of irreducible unitary Πu • representations of each reductive algebraic Lie describe Πq(G) (all representations); • group G. describe Πh(G) (hermitian representations) as a subset of (G); and Since we haven’t yet reached this goal, we want to Πq •describe (G) (unitary representations) as a identify steps that represent progress towards it. Πu subset of (G). Harish-Chandra in the 1950s, following a sugges- Πh tion of Chevalley, began to study the larger class of The first two steps of this program have been irreducible representations that are not necessarily addressed by Langlands and by Knapp-Zuckerman; unitary. A representation of a topological group there is an excellent account in Knapp’s book G means a continuous action of G on a complete Representation Theory of Semisimple Groups. Here locally convex topological vector space. We may is an approximate statement. write it as a “Theorem” 1 (Langlands, Knapp-Zuckerman). Suppose G is a reductive algebraic Lie group. Then π : G → Aut(Vπ ), the set Πq(G) of equivalence classes of irreducible with Vπ the vector space. We say that π is irreducible quasisimple representations of G is in natural if Vπ has exactly two closed invariant subspaces. bijection with a countable discrete collection of The study of irreducible representations in general complex algebraic varieties Xi (C). Each of these is complicated by the existence of invertible linear algebraic varieties is defined over R, and the subset operators on infinite-dimensional Banach spaces Πh(G) corresponds to the real points Xi (R). having no nontrivial closed invariant subspaces. Stated in this way, the “Theorem” is equally true Such operators define irreducible representations for reductive algebraic groups over arbitrary local of the group Z, but they have little to do with most fields. The quotation marks correspond to some problems of harmonic analysis. Harish-Chandra small and well-understood technical difficulties; for found a natural technical condition called quasisim- the experts, the magic words are “R-groups”. Each plicity on irreducible representations of reductive subset Πu(G) ∩ Xi (R) is defined by real algebraic Lie groups that excludes such pathological behav- inequalities; the difficulty is that we do not know a ior. (The condition is that all operators commuting simple description of those inequalities. with the representation are assumed to be scalar. In the case of unitary irreducible representations, Representations of Compact Lie Groups quasisimplicity is the theorem called Schur’s Lem- What is the nature of the information provided ma.) If G is any reductive algebraic Lie group, we by “Theorem” 1 above? The example of compact define Lie groups (which is a special case!) is helpful. I Πq(G) = equivalence classes of irreducible will go into some detail about that classical theory, quasisimple representations of G. seeking to formulate results in a way that carries

1024 Notices of the AMS Volume 54, Number 9 over to general reductive algebraic Lie groups. because the set P is very easy to compute and to An irreducible representation of a compact Lie manipulate. It is very unsatisfactory as a descrip- group is automatically quasisimple, Hermitian, and tion of the irreducible representations, because unitary; so those distinctions will not appear at all. it does not even explicitly describe the “natural Suppose K is a compact connected Lie group, bijection”. In to do that, we need a bit more and T is a in K (a maximal connected structure theory. Attached to each coroot α∨ there abelian ). Necessarily T is isomorphic to is a root α ∈ Πu(T ); the set of all roots is written a product of ` circles, where ` is a nonnegative R(K, T ) ⊂ u(T ). called the rank of K. Π Any irreducible unitary representation of T must The roots are by definition the nontrivial repre- be one-dimensional. A one-dimensional unitary sentations of T appearing in the “adjoint action” representation (of any topological group G) is a of T on the complexified Lie algebra of K. Cor- ∨ continuous homomorphism from G to the group responding to the simple coroots S are simple U(1) of 1 × 1 unitary matrices, which is again roots just the . Any homomorphism from S = {α1, . . . αm} ⊂ R(K, T ) ⊂ Πu(T ). (C) the circle to itself is given by raising to the mth Here is a description of the Cartan-Weyl bijection. power for some integer m. That is, Πu(circle) ' Z. Because T is a product of ` circles, it follows that Theorem 3 (Cartan and Weyl). In the setting of ` Theorem 2, an irreducible representation π of K u(T ) ' Z . In a coordinate-free way, we say that Π corresponds to a dominant weight µ ∈ P if and Πu(T ) is a of rank `. The structure theory of compact Lie groups only if the following conditions are satisfied: a) the provides a finite collection R∨(K, T ) of nonzero weight µ appears in the restriction of π to T ; and b) ∨ for every simple root α ∈ S, the weight µ + α does Z-linear maps α : Πu(T ) → Z, called the coroots of T in K. The structure theory points also to not appear in the restriction of π to T . certain natural subsets of the coroots, called simple Theorem 3 is in some sense a complete descrip- coroots; we fix such a subset tion of the irreducible representations of K, but it is still not completely satisfactory. It does not say ∨ = { ∨ ∨ } S α1 , . . . αm . (A) how to calculate the of a representation The simple coroots are linearly independent, so or its restriction to a compact subgroup of K. We they define a nonempty cone will address those questions (and generalizations ∨ ∨ ∨ for noncompact groups) in the next section. P = {µ ∈ Πu(T ) | α (µ) ≥ 0 (α ∈ S )}, (B) called the cone of dominant weights. Character Tables for Lie Groups It is a fundamental fact that everything about Much of the content of the Atlas of finite groups the structure of the compact Lie group K—and and representations consists of character tables. indeed of arbitrary reductive algebraic groups—is In this section I’ll recall what that means, how encoded by the lattice Πu(T ) and the finite set to extend the notion to Lie groups, and how it’s ∨ of Z-linear maps S . (To be precise, one needs possible to write character tables for reductive Lie also the simple roots S ⊂ Πu(T ), introduced in (C) groups in a finite form. below.) In the hands of Chevalley and Grothendieck Suppose π : G → Aut(Vπ ) is a representation and others, this fact led to the theory of reductive of a topological group on a finite-dimensional groups over arbitrary fields (or even commutative complex vector space Vπ . The character of π is a rings). For the atlas project, it means that the complex-valued function on G, defined by structure of reductive groups can be described in terms of strings of and so is perfectly Θπ (g) = tr π(g). suited to exact computer calculation. It’s very easy to see that Θπ is a on G Theorem 2 (Cartan and Weyl). Suppose K is a com- (that is, Θπ is constant on conjugacy classes in G). pact connected Lie group, and T is a maximal torus What is not quite so obvious, but still elementary, is in K. Pick a set S of simple coroots for T in K as in that irreducible (finite-dimensional) representations (A) above, and define dominant weights P as in (B). having the same character are equivalent. Then there is a natural bijection between dominant In the case of a finite group G of order N, the weights and irreducible representations of K eigenvalues of G are Nth roots of unity, so the values of characters of G are integer combinations P ↔ (K). Πu of the N complex numbers exp(2πmi/N) (for m In the language of the Langlands and Knapp- an integer between 0 and N − 1). It is therefore Zuckerman Theorem 1, P parametrizes the count- possible to write a character of G precisely: for able set of algebraic varieties Xi . Each Xi consists each in G, one can write the N of a single point, so Xi (C) = Xi (R). integers that are the coefficients of these roots This theorem is very satisfactory as a parame- of unity. (Fortunately it is possible in practice trization of the irreducible representations of K, to find far more compact representations of the

October 2007 Notices of the AMS 1025 character values.) A for G is a list continuous function f on M defines a generalized of all the character values of all the irreducible function by the formula representations. Z If G is a compact Lie group, the notion of char- f (ξ) = f (m)dξ(m). M acter table still makes sense (since the irreducible The of a finite-dimensional representation representations are finite-dimensional). What is of a Lie group G is a continuous function on G not so clear is whether it can be written down in a and therefore may be regarded as a generalized finite way. The values of each character must be Θπ function. The following theorem of Harish-Chandra specified at each of the infinitely many conjugacy shows that the character of an irreducible qua- classes in G; and then this task must be repeated sisimple representation of a reductive algebraic Lie for each of the infinitely many π. group G is a generalized function on G. solved these problems. I’ll write his solution completely in the simplest case and Theorem 5 (Harish-Chandra). Suppose G is a then say a few words about the general case. reductive algebraic Lie group, π is an irreducible quasisimple representation of G on a Hilbert Theorem 4 (Weyl). Suppose space, and ξ is a test density on G. The operator G = {a + bi + cj + dk | a2 + b2 + c2 + d2 = 1} R π(ξ) = G π(g)dξ(g) is trace class, and defining is the group of unit , and Θπ (ξ) = tr π(ξ) makes Θπ a generalized function on G. = { + | 2 + 2 = } = { | ∈ } ⊂ T a bi a b 1 exp(iθ) θ R G There is a conjugation-invariant open subset is a maximal torus. a) There is exactly one irre- G0 ⊂ G, whose complement has measure zero, so 0 ducible representation πn of G for each strictly pos- that the restriction of Θπ to G is a conjugation- 0 itive integer n. b) Every conjugacy class in G meets invariant analytic function Θπ , locally integrable T , so a class function on G is determined by its re- on G. The generalized function Θπ is equal to 0 striction to T . c) The value of the character of πn integration against Θπ . = on T is Θπn (exp(iθ)) sin(nθ)/ sin(θ). Writing a character table for the reductive alge- We have in (c) an infinite character table pre- braic Lie group G means writing down each of the 0 sented in finite form. The infinitely many rows are functions Θπ , as π runs over the (infinite) family indexed by n, and the infinitely many columns by of irreducible quasisimple representations of G. θ (more precisely, by θ up to sign and addition of The reason that each such function can be written multiples of 2π). down is that it turns out (just as in the case of In part (a) of the Theorem, one can think of the compact groups) to be a quotient of finite integer integer n−1 as corresponding to a one-dimensional combinations of exponential functions. representation of T , that is, to an element of Πu(T ). The possibility of handling infinitely many π is A version of (a) for general compact Lie groups a consequence of the Jantzen-Zuckerman “transla- is provided by the Cartan-Weyl Theorem 2 in the tion principle”. They partition all irreducible repre- last section. Part (b) makes sense as stated for sentations into finitely many translation families. In a general compact connected Lie group and is the formulas for the characters of representations true. The for general G in one translation family, only the exponential looks something like (c). There is a denomina- functions change: the coefficients in the formulas tor (generalizing the function sin(θ)) that is a remain the same. In the case of a , trigonometric polynomial on T , independent of there is single translation family, with characters the representation. The numerator (generalizing given by the Weyl character formula; all that varies sin(nθ)) is a trigonometric polynomial built from with the representation are the exponents. For the weight µ that parametrizes the representation. example, for the group described in For infinite-dimensional representations, the Theorem 4, the parameter for the translation family difficulties with are more funda- is the integer n. mental. The operators π(g) are essentially never Here is a little more detail. The differential of trace class. Harish-Chandra understood that the equations for the character Θπ come from the character makes sense only after “regularization” center Z of the universal enveloping algebra of the in the sense of distribution theory. Each individual Lie algebra of G. They are eigenvalue equations; operator π(g) does not have a trace: one first has the eigenvalues are the complex scalars by which to smooth the operator by averaging over a nice Z acts in the representation π. (That Z does act compact collection of values of g. by scalars is exactly Harish-Chandra’s definition Here is how to do that. Recall that a test density of quasisimplicity.) The eigenvalues are encoded on a smooth manifold M is a compactly supported by an algebra homomorphism λ: Z → C called the complex-valued measure ξ on M, which in local infinitesimal character of π. coordinates is a smooth multiple of Lebesgue mea- After appropriate (very subtle!) changes of sure. A generalized function on M is a continuous variables, the differential equations become (in linear functional on the space of test densities. Any local coordinates) systems of constant-coefficient

1026 Notices of the AMS Volume 54, Number 9 eigenvalue equations on Rn. For each choice λ of Theorem 6 (Langlands and Knapp-Zuckerman). eigenvalues, the solutions are finite linear combina- There is a natural bijection between the set {π} tions of exponential functions. Harish-Chandra was of irreducible representations of G and Harish- able to describe these solutions very explicitly and Chandra’s solutions (D) to the differential equations completely. (Much of the difficulty is understanding for characters. Write πi for the irreducible repre- how solutions on different coordinate patches fit sentation corresponding to the solution Θi . Then together.) For each choice of eigenvalues, he found the (square) matrix P πi ,j in (E) above is a lower an explicit basis triangular integer matrix with 1s on the diagonal. To say that the matrix is lower triangular re- , ,..., (D) Θ1 Θ2 ΘN quires an appropriate ordering of the solutions for the global solutions of the differential equations. Θj . Harish-Chandra’s construction of the solutions (It would be mathematically more precise but nota- analyzes exponential terms of greatest possible λ λ λ growth at infinity on G. If we assume that the are tionally more burdensome to write Θ1 , Θ2 ,..., ΘN(λ), Θj indicating explicitly the fact that the equations ordered so that the later ones have faster growth, being solved depend on the system λ of eigenvalues. then we get the lower triangularity. Another way I will choose the path of unburdened imprecision.) to achieve it is explained after equation (F) below. In this section I will say a bit about the mathemat- As a consequence, each irreducible character Θπ has a unique expression ics underlying the computation of the matrix P πi ,j . The main tool is a geometric reinterpretation of N X the matrix introduced by Beilinson and Bernstein. = P , (E) Θπ π,j Θj It takes place in a smooth complex projective j=1 algebraic variety X (depending on the reductive for some complex numbers P π,j . (The reason for group G) called the complete flag variety X = X(G). calling the coefficients P will emerge in the next One way to define X is as the variety of maximal section: the overline means the in a quotient solvable Lie subalgebras inside the complexified map, defined from polynomials to coefficients Lie algebra gC of G. by evaluation at 1.) Writing down an irreducible The variety X is degenerate in some very in- character Θπ is therefore equivalent to writing teresting ways. Most algebraic varieties are not 1 down the N complex numbers P π,j . P bundles in any way. The variety X has a finite The last important fact is that the coefficients collection P π,j are all integers. It is not quite clear to whom πs : X → Xs (s ∈ S) (F) this observation should be attributed. At least with 1 forty years of hindsight, it is easy to deduce from of P fibrations. (That is, each πs is a smooth 1 the work of Langlands and Knapp-Zuckerman on submersion with fiber the Riemann sphere CP .) the classification of irreducible representations. The parametrizing set S is the set of simple roots Zuckerman may have been the first to recognize the introduced in (C) above. existence and importance of character formulas In case G is GL(n, R), the variety X may be identified with complete flags in n: increasing (E) with integer coefficients. He wrote an explicit C chains of n linear subspaces F , with dim F = j. formula for the character of a finite-dimensional j j There are n − 1 P1 fibrations; for 1 ≤ j < n, the jth representation (φ, F) in his thesis; in that case the fibration arises by throwing away the j-dimensional coefficients P φ,j are all ±1 or zero. subspace Fj in a complete flag. The ideas of Beilinson and Bernstein concern How to Compute the Characters the equivariant geometry of X. One might expect Equation (E) above says that the character table that what ought to matter is the action of G on X. for a real reductive Lie group G may be ex- For technical reasons, however, what enters their λ pressed as a matrix of integers P π,j ; here I have work is equivariance with respect to K(C), the temporarily reinserted the dependence on the complexification of a maximal compact subgroup infinitesimal character λ. The index j runs over K ⊂ G. Harish-Chandra’s solutions (D) to the differential Theorem 7 (Beilinson and Bernstein) Suppose G is equations (with fixed eigenvalues λ). The index π a real with complete flag variety X, runs over irreducible representations of G (with K is a maximal compact subgroup of G, and K(C) fixed infinitesimal character λ). is its complexification (an acting on The Jantzen-Zuckerman translation principle X). says (in a very explicit and computable way) that, as a) Harish-Chandra’s solutions (D) to the differ- λ varies, there are only finitely many possibilities λ ential equations are naturally in one-to-one corre- for P π,j . Henceforth I will therefore drop the λ and spondence with pairs (Zo, L) consisting of a K(C) speak only of computing one matrix P π,j . Thinking orbit Zo on X and a K(C)-equivariant local system about this matrix is simplified by L on Zo.

October 2007 Notices of the AMS 1027 According to Theorem 6, exactly the same parame- topological invariant of the singular algebraic vari-

ters index the irreducible representations of G. ety Zi . It measures at the same time the nature of

b) Suppose πi is an irreducible representation, the singularity of Zi (in the theorem, the singularity corresponding to the pair (Zi,o, Li ). Write Zi for the at points in Zj,o) and the possibility of extending closure of Zi,o, a (possibly singular) algebraic subva- the local system Li from the open subset Zi,o to all of Z . riety of X. Suppose Θj is one of Harish-Chandra’s i solutions (D), corresponding to the pair (Zj,o, Lj ). As the term “Euler characteristic” suggests, in- tersection homology provides (for each i and j) not Then the character formula coefficient P πi ,j of (E) is equal to the Euler characteristic of the local inter- just a single integer but rather a finite collection of m section homology of Zi with coefficients in the local nonnegative integers pij , the ranks of individual system Li , evaluated at (Zj,o, Lj ). local intersection homology groups. I will modify Because Harish-Chandra solved systems of the indexing of the homology in order to arrange that the index m can run from 0 to the complex differential equations to get the Θj , and Beilinson and Bernstein work with derived categories of codimension of Zj in Zi . (One of the key properties constructible sheaves, you may imagine that of intersection homology is that the top degree there is some work required to pass from what can appear only if Zj = Zi .) A consequence of Beilinson and Bernstein proved to the statement this reindexing is that the Euler characteristic of of Theorem 7. This translation was one of my own Theorem 7(b) is

contributions to the subject. The most difficult dim Zi −dim Zj X m m (−1) (−1) pij . part was convincing some of the experts that the m result was really not quite obvious. If Z is smooth and L is the trivial local system, Despite my promise at the beginning to be his- i i then pm is equal to zero unless m = 0, Z is torically incomplete, I have been asked to interrupt ij j contained in Z , and the local system L is also the mathematics here to say a few words about the i j trivial; in that case p0 is equal to 1. sources of the ideas in Theorem 7. Kazhdan and ij The Kazhdan-Lusztig polynomial for the pair Lusztig in 1979 formulated a precise conjecture (i, j) is by definition describing the characters of irreducible highest weight modules in terms of the combinatorics X m m/2 ,j (q) = pij q . of the . They observed that the first m appearance in these characters of coefficients The parameters i and j represent local systems on ± other than 1 can be empirically related to the orbits of K(C) on the complete flag variety X. The first failures of local Poincaré duality for singular polynomial can be nonzero only if the orbit Zj,o Schubert varieties. MacPherson suggested that is contained in the closure of the orbit Z ; this their observations might be formalized using inter- i,o explains the “lower triangular” result in Theorem section homology. Kazhdan and Lusztig did this, 6. It turns out that the local groups vanish in odd proving that their combinatorial construction in degrees, so that P is actually a polynomial in q the Weyl group actually calculated the intersection i,j (with nonnegative integer coefficients). (This is a homology of Schubert varieties. This calculation special fact about the varieties Z , not a general fact is short but very deep, using the tools developed i about intersection homology.) The degree of P is by Deligne to prove the Weil conjectures. It seems i,j bounded by half the complex codimension of Z to have been an inspiration for the development j in Z ; the bound is strict if i 6= j. (This is a general by Beilinson, Bernstein, Deligne, and Gabber of i fact about intersection homology.) Because of the the general theory of perverse sheaves, which has vanishing in odd degrees, the Euler characteristic since become a basic tool in representation theory is just the (nonnegative) value at q = 1, times the and algebraic geometry. sign (−1)dim Zi −dim Zj . The Kazhdan-Lusztig conjectural character for- The point of Theorem 7 is that characters can be mula (still for highest weight modules) was now a statement involving intersection homology of computed from Kazhdan-Lusztig polynomials and Schubert varieties. In this form it was proved inde- that these polynomials depend on the geometry of pendently by Brylinski-Kashiwara and by Beilinson- K(C) orbit closures on the complete flag variety Bernstein, using the algebraic theory of differential X. The next theorem describes the geometric tools equations created by Sato, Kashiwara-Kawai, and needed to compute intersection homology for these Beilinson-Bernstein. orbit closures. I will not try to describe “intersection homol- Theorem 8 (Wolf). Suppose G is a real reductive ogy” here. (The book Introduction to Intersection group with complete flag variety X, K is a maximal Homology Theory by Kirwan and Woolf is highly compact subgroup of G, and K(C) is its complexifi- recommended by my colleagues who should know.) cation (an algebraic group acting on X). In order to understand the nature of the statement, a) The action of K(C) on X has finitely many what matters is that intersection homology is a orbits.

1028 Notices of the AMS Volume 54, Number 9 Suppose Zo is an orbit of K(C) on X, and s ∈ S. local systems. Keeping track of these local systems Write Z for the closure of Zo (a projective algebraic under the maps πs is subtle and was another of subvariety of X). Define Zs = πs (Z) (cf. (F) above), a my contributions to this mathematics. (One ends projective algebraic subvariety of Xs , which we call up in some cases with inductive formulas not for s −1 the s-flattening of Z. Define Z = πs (Zs ), which individual Kazhdan-Lusztig polynomials, but for we call the s-thickening of Z. The map πs exhibits sums of two polynomials, corresponding to two s 1 Z as a P bundle over Zs . local systems. Part of the difficulty is to find a way b) The s-thickening Zs is the closure of a unique to solve the resulting collection of equations.) s K(C) orbit Zo. A critical point is that the algorithm needs to c) There are two mutually exclusive possibilities. know the highest degree coefficients and not just 1) The s-thickening Zs is equal to Z, so that Z is the values of the polynomials at q = 1. Even though 1 s a P bundle over Zs . In this case Zo = Zo. 2) The we are interested (for character theory) only in map πs from Z to Zs is generically finite (and in values of the polynomials at 1, the algorithm does particular is finite over the orbit Zo). In this case not allow us to compute only values at 1. s s dim Zo = dim Zo + 1; and Zs = (Z )s . In particular, Once the algorithm has forced us to look at s the thickened orbit Zo falls in case (1). coefficients of the Kazhdan-Lusztig polynomials, it d) Every orbit of K(C) on X arises by a finite is very natural to ask for representation-theoretic succession of thickening operations applied to some interpretations of those coefficients. There is a closed orbit. great deal to say on this subject. I will mention Part (a) of this theorem is due to Wolf. The only that the top degree coefficients mentioned 1 remaining assertions are quite easy, although it is above turn out to be dimensions of Ext groups more difficult (for me at least) to attribute them between irreducible representations. precisely. The idea of constructing and describing the orbits in this way has its roots in the theory of The Atlas of Lie Groups and Schubert varieties and so is very old. Representations Theorem 8 describes the geometry of orbit That brings the mathematical story up to about closures and so leads to Kazhdan and Lusztig’s 1985. I’ll now turn away from abstract mathematics, algorithm for computing intersection homology by toward the story of the atlas project. induction on the dimension. Here is a sketch. An In 2002, Jeff Adams had the idea of getting orbit of minimal dimension is closed and is itself a computers to make interesting calculations about complete flag variety for the algebraic group K(C). infinite-dimensional representations of reductive Such varieties are smooth, so the local intersection Lie groups: ultimately, he hoped, to calculate uni- is simple. According to part (d), any tary representations. Of course as mathematicians orbit Wo of greater than minimal dimension must we want completely general theorems, and it’s by s arise by thickening: Wo = Zo, with Zo an orbit of no means clear that there is a finite calculation to dimension one less. Now the orbit closure Z is a find the unitary duals of all reductive groups at ramified cover of the flattening Zs (according to once. But the work of Dan Barbasch (for example, (c)(2)); so the intersection homology of Zs is very his classification of the unitary duals of the com- close to that of Z, which is known by induction. plex classical groups) makes it possible to hope Finally the orbit closure W is a P1 bundle over that one can find a finite description of the unitary Ws = Zs ; so the intersection homology of W is duals of all classical Lie groups. The exceptional made in a simple way from that of the flattened groups are finite in number, so treating them is variety Zs . a finite calculation. That became Jeff’s standard Making this sketch precise uses versions of the for measuring the effectiveness of any piece of Weil conjectures for intersection homology, proved representation-theoretic software: could it treat by Beilinson, Bernstein, and Deligne. The most the largest exceptional group E8? subtle point is descent from Z to Zs by the ramified It was clear that the first problem was to write covering map πs . What happens there is that the a program that could work with the Cartan sub- intersection homology down on the flattening Zs groups, maximal compact subgroups, and Weyl arises from that on the covering Z by removing groups of any real reductive group. Jeff recruited something. Exactly what should be removed is Fokko du Cloux to do this, and Fokko began to work determined by certain highest degree intersection in 2003. By 2004 his software could handle this homology of Z; that is, by top degree coefficients structure theory (better than most mathematicians, in Kazhdan-Lusztig polynomials. at least). In the setting of Kazhdan and Lusztig’s original The next step was less obvious, but Fokko work, the orbits Zo are simply connected, so the and Jeff settled on computing Kazhdan-Lusztig local systems involved are all trivial. For general polynomials for real groups. Fokko had written the real groups the orbits have fundamental groups best software in the world to do this for Coxeter of size up to (Z/2Z)rank and therefore a wealth of groups; the algorithms for real groups are similar

October 2007 Notices of the AMS 1029 in structure (although much more complicated Among the exceptional groups, that left the split in detail, because of the complications attached form of E8. to local systems). Theorem 7 above says that knowing these polynomials provides formulas for Warming Up for E8 irreducible characters; it is also a critical step in How big a computation is the character table for several computational approaches to classifying split E8? Fokko’s software told us that there were unitary representations. exactly 453,060 (translation families of) irreducible So late in 2004, Fokko began to add to his soft- representations. According to Theorem 6 above, ware an implementation of the Kazhdan-Lusztig the character table can be described by a square algorithm for real groups. The papers in which matrix of integers, of size 453,060. The number of this algorithm is formulated are extremely dense, entries is therefore about 2 × 1011, or 200 billion. and written with no consideration for computa- Fortunately the matrix is lower triangular, so we tional practice. An expert could easily spend many only need 100 billion entries. months just to understand the mathematical state- Unfortunately we need to calculate not the ments. Fortunately, Jeff Adams had been working entries directly, but rather the Kazhdan-Lusztig on a new formulation of Theorem 6 (parametrizing polynomials whose values at 1 are the entries. The irreducible representations), growing out of earlier degrees of the polynomials are bounded by 31; work that he did with Dan Barbasch and me. Jeff’s we expected an average degree of about 20, and formulation seemed suited to computer implemen- therefore a total number of coefficients around 2 tation; he had been working with Fokko to make it trillion. more so. Fortunately many of the matrix entries are easily Over the course of the next year, Fokko un- seen to be equal. An example is Zuckerman’s derstood the Kazhdan-Lusztig algorithm for real formula for the character of a finite-dimensional groups, recasting it in the language that he and Jeff representation, where I said that all the coefficients had developed. He wrote clear and efficient code are ±1 or 0; this is the last row of the character to implement it. In November of 2005—incredibly matrix. In the case of E8, there are 320,206 nonzero soon!—he finished. Very quickly he and Jeff used terms in this row. Fokko’s software recognizes that the software to compute Kazhdan-Lusztig polyno- all 320,206 of those Kazhdan-Lusztig polynomials mials (and so character tables) for all of the real are going to be equal and stores only the diagonal forms of F , E , and E , and for the non-split form 4 6 7 entry 1. In general one needs to store only one of E . 8 representative polynomial for each family of “obvi- The most complicated of these calculations is for ously equal” entries. Fokko’s software calculated the non-split form of E . There are 73,410 distinct 8 how many such families there were: a bit more than (translation families of) irreducible representations, 6 billion. So we were down to storing about 6 billion so the character table is a 73, 410 × 73, 410 matrix polynomials with about 120 billion coefficients. of integers. The integers are values at q = 1 of Unfortunately we had no clear idea how big the Kazhdan-Lusztig polynomials. These polynomials (nonnegative integer) coefficients of these polyno- have degrees from 0 to 27. Their coefficients are mials could be. In the case of split D , the largest nonnegative integers, of which the largest is 2545. 5 is 5. For split E , the largest is 27, and for split The total number of distinct polynomials appearing 6 E , it’s 3583. This trend was not encouraging; it (among the three billion or so entries below the 7 seemed clear that the coefficients would exceed diagonal in the matrix) is 10,147,581. Here is the 65, 535 = 216 − 1, so that they could not be stored polynomial with largest coefficient: in two bytes (sixteen bits) of computer memory. q13 + 30q12 + 190q11 The next practical size is four bytes. +682q10 + 1547q9 + 2364q8 Fortunately Fokko wrote the software to com- pute with four-byte integers and to test carefully +2545q7 + 2031q6 + 1237q5 for numeric overflow throughout the computation. 4 3 2 +585q + 216q + 60q + 11q + 1 If overflow happened, the plan was to switch to It’s hard to say what constitutes a “typical” eight-byte integers and try again. polynomial, but here is the one at the midpoint of Unfortunately, 120 billion 4-byte integers re- the lexicographically ordered list: quire 480 billion bytes of RAM, or 480G. That’s a q9 + 7q8 + 13q7 + 6q6 + 6q5 lot of RAM. (The nature of the Kazhdan-Lusztig algorithm, which constantly looks at widely dis- 4 3 2 +14q + 18q + 16q + 7q + 1. tributed results from earlier in the computation, Fokko’s software will calculate this character table makes storing results on disk impractically slow. on my laptop in about half an hour, using 1500 We tried!) megabytes of RAM. (I bought a big memory chip Fortunately, some of the six billion polynomials at about the same time as I sold my soul to the are zero, and some of them are equal to others silicon devil.) “by chance” (that is, for reasons that we have yet

1030 Notices of the AMS Volume 54, Number 9 to understand). So Fokko wrote the software to (Not) Buying a Really Big Computer store only one copy of each distinct polynomial. Birne Binegar and Jeff Adams suggested that we He hoped that the number of distinct polynomials start looking seriously at applying for an NSF grant might be a few hundred million: so perhaps 6 billion to buy a machine with perhaps 256G of RAM: coefficients, requiring 25G of RAM. The indexes something that might cost US$150,000 or more. I keeping track of all these polynomials would also asked a number of mathematicians whether they occupy a lot of memory: by the most optimistic might be able to make use of such a computer. estimates perhaps 45G, but quite possibly a lot Noam Elkies had a fascinating reply. First he ex- more. plained some theoretical limits on the computations Unfortunately, we didn’t have a computer with that could use a lot of memory. even 50G of RAM. A computation that actually uses Fortunately computer science is often computer N bytes of storage must take time art, and even Fokko’s work could be improved. at least N. But once N gets as large as 256GB it might not be feasible Fokko worked on that constantly during 2006, to spend much more than N time: and Marc van Leeuwen began to make serious N·log(N) or N·log2(N) is certainly contributions as well. The two of them rearranged OK (e.g., fast integer or polynomial the indexes and the code in some extraordinarily arithmetic, and other applications clever ways. of the Fast Fourier Transform; also Unfortunately, tests running partway through solving f (x) = f (x0) by sorting N the E8 calculation (done mostly by Birne Binegar) values of f (x) and finding a consec- revealed that Fokko’s first hopes about the number utive match); maybe also N3/2 (e.g., of distinct polynomials were too optimistic. Even an linear algebra with dense matrices optimistic reading of Birne’s tests suggested more of size N1/2, or computing the first like 800 million distinct polynomials, meaning N coefficients of modular forms perhaps 60G or more to hold the coefficients. such as ∆ without fast arithmetic); Fortunately Dan Barbasch is now chair of the probably not N2. So there might not math department at Cornell, and in September of be all that much room for making 2006 he managed to gain access to a machine with use of such a huge machine… 128G of RAM and 128G of swap space. He used it He went on to ask to run the E computation to the end. The fact that 8 Is it clear that the E8 computation Fokko’s overflow tests were not set off showed cannot fit into “only” 128 or 64GB? that all the coefficients really fit in four bytes. I explained the demands of polynomial storage: Unfortunately he had no reasonable way to We know that the polynomial co- write the results to disk, so they disappeared. efficients can exceed 216 (by com- (Fokko’s software was written to produce output in putation), and we hope that they human-readable form. In the case of E8, his output don’t exceed 232. Each polynomial for the character table would have consisted of is stored as a vector of 32-bit in- about fifty billion lines (one for each nonzero tegers, of size exactly equal to its entry in the character table) averaging about 80 degree plus one. Assuming an av- characters. As a disk file this would have been erage degree of 19, that’s 80 bytes several terabytes.) Also unfortunately, Dan didn’t per polynomial. have the improvements that Fokko and Marc had On November 30, Noam replied made to the code: Dan’s computation used 224G Well 232 is less than the prod- of memory (half of it swap space). Because of the uct of the ten odd primes less use of swap space, it took twelve days to finish. than 25, so unless the computation Fortunately, by November of 2006, Fokko and requires divisions by numbers oth- Marc had trimmed memory use in the code a great er than 2 you could reduce this deal. Through the persistence of Birne Binegar, and from 80 bytes to something like the generosity of number theorist William Stein, the (5/32)·80 = 12.5 bytes, at the cost atlas group got access to William Stein’s computer of running the computation 9 times sage at the University of Washington (with 64G of (counting once for mod 3 · 5). RAM and 75G of swap). On this machine we could In other words, we needed to stop thinking about finally do some large fraction of the E8 character intersection cohomology for a while and use the table computation. By late November, we believed Chinese Remainder Theorem. Noam’s suggestion that we could finish E8 with about 150G of RAM. was to compute the Kazhdan-Lusztig polynomials Unfortunately, 150G is just a little more than modulo m for several small values of m and sage has, even with swap. to store the results to disk. A second program

October 2007 Notices of the AMS 1031 could (for each polynomial) read in these several Marc fixed things instantly, and on Thursday mod m reductions; apply the Chinese Remainder evening December 21 we started a calculation Theorem to compute the polynomial modulo the mod 256 on sage. This computed 452,174 out of least common multiple of all the moduli; and write 453,060 rows of the character table in 14 hours, the result to disk. This second program would need then sage crashed. We tried again starting late to have only a handful of polynomials in RAM at Friday afternoon, and actually finished with good the same time, so it could run on a much smaller output: the character table mod 256 was written to computer. disk! Because we used multi-threading to speed up I started to compose a reply explaining why the computation, this run took just eleven hours. modular reduction of the Kazhdan-Lusztig algo- On Saturday December 23 we started a calcu- rithm doesn’t work, but I had to throw it away: lation mod 255. This time sage crashed a third the algorithm works perfectly over Z/mZ. Fokko’s of the way through the computation. There was code was beautifully compartmentalized, and Marc no one physically present to reboot it (apparently van Leeuwen is amazing, so by December 4 we were some kind of holiday in Seattle) so we retired for a getting character table entries mod m for any m up bit (still having mod 256 as our only good output to 256. In these calculations, we needed just one files). byte of memory for each polynomial coefficient. A Meanwhile Marc van Leeuwen had written code billion polynomials of degree 20 could live in 20G to combine Kazhdan-Lusztig polynomials from of RAM. several moduli m1, m2,... into Kazhdan-Lusztig polynomials modulo lcm(m1, m2, . . .). We tested Computing Characters Mod m the code on the first hundred million entries of On December 6 Marc’s modifications of Fokko’s the E8 character table modulo 253, 255, and 256 code on sage computed about three-fourths of (which we could calculate on smaller computers the entries in the E8 character table mod 251, than sage), and it worked fine. When sage came finding almost 700 million distinct polynomials back up on December 26, we got a character table and using 108G of memory. Since 90% of that mod 255 written to disk. At 1 a.m. on December 27, was indexes, working on their structure became we started a run mod 253. About halfway through, worthwhile. Marc redesigned the indexes rather sage crashed. completely (replacing 8-byte pointers by 4-byte The experts I consulted assured me that the counters several billion times, for example). In the atlas software couldn’t possibly be crashing sage. end he reduced the size of the indexes to about My own opinions about the causes of the crashes 35G; they would have required more than 100G for wavered between black helicopters from the NSA the original code. He also added code to output and Sasquatch. We resolved to keep our hands off the answers to (small machine-readable) disk files. sage until we were older and wiser: say for a year. Meanwhile Birne Binegar ran various versions On Wednesday January 3 we were all one year of the code on sage. Among other things he older, which made perhaps thirty years of addi- established for certain that there were more than tional wisdom counting all the atlas people. This one billion distinct polynomials. factor of thirty seemed like a suitable margin Early on December 19, Marc’s modified code of safety, so that afternoon we started another began a computation for E8 mod 251, with the computation mod 253. This finished in twelve possibility of actually writing the result usefully hours. at the end. Essentially it worked, finishing the computation in about 17 hours. From diagnostic The Chinese Remainder Calculation output of the software, we learned that there were By 4 a.m. Thursday January 4th we had output exactly 1,181,642,979 distinct Kazhdan-Lusztig for three moduli (253, 255, and 256) with least polynomials mod 251. (That turned out to be the common multiple 16,515,840: bigger (we had some number over Z as well.) The calculation used only hope) than all the coefficients of the Kazhdan- 65G of memory; the improvement over 108G on Lusztig polynomials. Marc van Leeuwen took unfair December 6 was because of Marc’s redesigned advantage of the time difference in Europe to start indexing system. running his Chinese Remainder Theorem utility on But writing the answer to disk took two days. the results. Its first task was to correlate the indices Marc and I went over Marc’s output code to see why. of the three output files, to determine which (of We figured it out, and Marc improved the speed. 1.1 billion) polynomials mod 253 corresponded to But we found at the same time a bug: he wrote which mod 255. That finished in nine hours. size() in one line where he meant capacity(). At that point we encountered another speed The result was that, even though the polynomials problem. The first version of Marc’s software had a were all correctly written to disk, the index files counter displaying the number of the polynomial (explaining which polynomial was stored where) to which the Chinese Remainder Theorem was were missing something like half their contents. being applied, to allow for monitoring progress.

1032 Notices of the AMS Volume 54, Number 9 Since there are more than a billion polynomials, the beginning of this article.) I was convinced that this meant writing several billion characters to the we’d find larger coefficients among the 350 million display. It turns out that takes a (very) long time. So polynomials that were not correctly evaluated the we started over on Friday morning January 5, with a first time and that we’d need a larger modulus counter that updates only every 4096 polynomials. than 16,515,840 to get them all. Everything went nicely until sage crashed. So at 6 a.m. on Sunday January 7th I was able William Stein identified sage’s problem as a to restart Marc’s current (and correct) Chinese flaky hard drive. The situation for atlas was Remainder Theorem utility, this time adding mod- this: sitting on sage’s flaky hard drive were 100 ulus 251. Of course nothing went wrong (because gigabytes of output files, the Kazhdan-Lusztig what could go wrong?), and the last polynomial polynomials modulo 253, 255, and 256 for E8. was written to disk just before 9 a.m. Eastern time Each of these files represented something like on Monday January 8. twelve hours of (multi-threaded) computation; the hundreds of hours of unsuccessful computations What Next? made them feel a great deal more valuable. Sixty gigabytes is too much information to look William Stein replaced the bad hard drive with at, even for nineteen mathematicians working in a good one, on which he had made daily backups 1 seamless harmony. Here are some of the “clean of all the work on sage. He did this more or less and simple results” that I promised in the in- instantly: we still had our data files. I was already troduction. Attached to any representation are deeply indebted to his generosity in allowing the many beautiful geometric invariants. Knowledge atlas group access to sage, but this raised him of the character table allows us to compute some even higher in my esteem. of them. We have computed the Gelfand-Kirillov We restarted the Chinese Remainder calculation dimension of each irreducible representation of E . late Friday afternoon January 5. 8 This is an integer between 0 and 120 that mea- Early Saturday morning, the disk file of polyno- sures how infinite-dimensional the representation mial coefficients mod 16515840 had grown to be is. Finite-dimensional representations are those about 7 billion bytes larger than it was supposed of GK dimension 0, and generic representations to be. Since it was a day with a y in it, I assumed that Marc van Leeuwen would be working. I asked (in a technical sense coming from the theory of him to find out what was wrong. He was visiting automorphic forms) are those of GK dimension his family in the Netherlands for the weekend and 120. The finite-dimensional representations were had extremely limited access to the Internet. identified by Cartan and Weyl (Theorem 2 above) The bug was (to my eyes) unbelievably subtle. around 1930, and the generic representations by Since the number of polynomials is about a billion, Kostant in 1978 (in both cases for all real reductive Marc’s code represented the index of a polyno- groups). mial by a 4-byte integer (perfectly good up to Now we can say for E8 exactly what happens be- 4,294,967,295). At some point this integer needs to tween these extremes. For instance, of the 453,060 be multiplied by 5 (the number of bytes in the index (translation families of) representations we studied, entry for one polynomial); the result is put into an there are exactly 392 of GK dimension 57. We can 8-byte integer, where it fits nicely. But when the say which ones they are. (I chose 57 because it’s polynomial number exceeds 858,993,459, and the the smallest possible dimension of a nontrivial multiplication is done in four bytes, it overflows. homogeneous space Z57 for E8. These 392 families The result was that the code worked perfectly in of representations appear in sections of vector any reasonable test (like the one we ran with a bundles over Z57.) In the same way we can identify hundred million polynomials). James Arthur’s special representations, To complicate Marc’s task further, the bug was which conjecturally play a fundamental role in the not present in his most recent version of the code; theory of automorphic forms. There are 111 of what I was running on sage was a couple of days these among the 453,060 representations. older. In the longer term, our goal is to determine So he was looking for a subtle bug that wasn’t completely the unitary irreducible representations there, without Internet access to the machine where for the exceptional Lie groups. Our hope is that the problem occurred. It took him almost twenty knowledge of the character table will allow us to hours to find and fix it (here of course I assume that make a computation of these unitary representa- he neither slept nor ate nor spoke to his family). tions. Armed with a list of unitary representations, Marc’s analysis showed that the bug came into we can try to explain (most of) it using the Kirillov- play only around polynomial number 858 million; Kostant orbit method, or Langlands’ ideas about so all the coefficients (modulo 16,515,840) calcu- lated before that were correct. The largest of these 1This is a theoretical assertion. I have no practical experi- coefficients was 11,808,808, at polynomial number ence with a team of nineteen mathematicians working in 818,553,156. (That is the polynomial displayed at seamless harmony.

October 2007 Notices of the AMS 1033 functoriality, or perhaps even something entirely Members of the Atlas of Lie Groups and new. Representations Despite the length of this article, I have left out a great deal. I have said nothing about the meetings Jeffrey Adams of the atlas group, where the insights of all of Dan Barbasch the mathematicians involved contributed to the Birne Binegar shape of the software that Fokko was writing, and Bill Casselman to our evolving understanding of the mathematics Dan Ciubotaru beneath it. Fokko du Cloux The greatest omission is the personal story of Scott Crofts Fokko du Cloux. He was diagnosed with ALS (a Tatiana Howard progressive neurological disease) just after finish- Marc van Leeuwen ing the Kazhdan-Lusztig computation software Alfred Noel in November of 2005. By February 2006 he had Alessandra Pantano little use of his hands, and by May he was entirely Annegret Paul paralyzed below his neck. But he continued to Siddhartha Sahi share his skills and insights into the mathematics Susana Salamanca and the programming—and his great joy at meeting John Stembridge a formidable mathematical challenge—with Jeff Peter Trapa Adams and with Marc van Leeuwen and with me, David Vogan until his death on November 10, 2006. Wai-Ling Yee The atlas has introduced me to great mathe- Jiu-Kang Yu maticians I barely knew, like Marc van Leeuwen and We are very grateful to Brian Conrey and John Stembridge, and it has shown entirely new the American Institute of Mathematics, which depths in people I knew well, like Jeff Adams and provided financial support and a wonder- Dan Barbasch. So it’s a very high bar…but what has ful mathematical meeting place from the been best of all, mathematically and personally, beginning of the atlas project; and to the Na- has been spending time with Fokko. It’s still the tional Science Foundation, which has provided best: every minute I spend on this project is a support through FRG grant 0554278. chance to think about him, and that’s always good —D.V. for a smile. So thank you to Fokko, who did this all by himself. Thank you to everyone who helped him—Marc van Leeuwen did more of that than anybody, but there were a lot of indispensable people. I haven’t had a boss since I worked in a lum- beryard in the summer of 1972, until Jeff Adams. Thank you, boss! I hope to be back here when we have some unitary representations to share.

1034 Notices of the AMS Volume 54, Number 9