The Character Table for E8 David Vogan n January 8, 2007, just before 9 in What’s E8? the morning, a computer finished writ- A Lie group 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 Lie group: 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 Lie algebra. 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 vector space, 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 Wilhelm Killing 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, E6, E7, 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 real form, for which the cor- A general warning about mathematical precision. 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 basis 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 dimensions 14, 52, 78, 133, and 248) having no straightforward connection to Unitary Representations and Their classical geometry. 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 topological group 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 representation theory 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 function 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 direct sum. 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) = n2Z 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) = ξ2R C · e dξ given by the rest of this section, but nothing insurmountable.) Fourier transform. 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—subgroups of n × n space structure.