Computing in exceptional groups by Bill Casselman for Tom Hales’ 60th, University of Pittsburgh, June 2018 These slides can be found at http://www.math.ubc.ca/~cass/slides/hales-bday.pdf
! This is potentially a very dull and exceedingly technical topic. I should say that it is an improved version of material I sent to Tom in email a few years ago, and which I believe he found useful. I hope to exhinit some interesting problems that deserve to be better known. Some have already been solved, but there are also some that have not yet been finished off. My main, if somewhat eccentric, claim is that we do not yet really under- stand the structure of semi-simple groups, or even semi-simple Lie alge- bras. Whether this is important or not remains to be seen, but it seems to me that sometimes these old questions come close to main stream prob- lems. 2/38
Contents 1. Introduction ................................................................ 4 2. Chevalley’s contribution ................................................. 8 3. Primary references ...................................................... 21 4. Kottwitz’ contribution ................................................... 22 5. Tits’ contribution ......................................................... 30 6. What about the group? ................................................. 36 7. More references .......................................................... 38 3/38
1. Introduction 4/38
A complex Lie group is called reductive if every continuous finite-dimensional representation decomposes into a direct sum of irreducible ones. Some reductive groups are defined very simply in terms of matrices. For example the group SL n ( C ) which is the group of all n × n complex matri- ces of determinant 1 . Or the complex symplectic group Sp 2 n ( C ) of all of all 2 n × 2 n complex matrices X such that � � �� 0 − I t XJX = J J = . I 0 But in the late nineteenth century a finite number of exceptional reductive groups were found to be characterized in more abstract terms. For exam- ple, the group G 2 of dimension 14 , which is rather sparsely embedded in GL 7 ( C ) . There are altogether five exceptional types, and for each of them certain useful matrix realizations are known, but these are difficult to work with. If I understand correctly, the classification of their Lie algebras came first. Although the terminology was different, this was in terms of what we now call root systems • . 5/38
It seems to have been L. E. Dickson who first realized that nearly all of these complex matrix groups should have analogues defined over finite fields. In particular, in a real tour de force he managed to show that there was an analogue of the smallest exceptional group, now called type G 2 . He constructed it in terms of its embedding into GL 7 , and the size of G ( F q ) is q 6 ( q 6 − 1)( q 2 − 1) . The other four exceptional groups remained a mystery. Dickson had much more trouble with fields F q when q = 2 n . It was in- evitable that in his approach small finite fields will cause difficulties. How to distinguish Sp from SO ? This line of inquiry was initiated by Galois, looking for simple groups. 6/38
The situation was completely changed about 1955 when Chevalley discov- ered how to deal uniformly with reductive groups over any field, in terms of what is now called root data, a variant of the notion of root system. Chevalley’s discovery amounted to a revolution. His discovery is many years behind us, and much of the subject has by now become very familiar. There are parts of his work, however, which are now almost forgotten—partly because its more technical aspects are not necessary to work productively in the field. These aspects have be- come part of the machinery behind the curtain, so to speak. But with the possibility of using computers to carry out computations in arbitrary reductive groups, questions raised by Chevalley have come into light again, and that’s what I’ll talk about. I have to confess, however, that I do not know if this material will ever be- come part of the main stream of the subject. 7/38
2. Chevalley’s contribution 8/38
Let G be a split reductive group defined over any field F in which one knows how to do arithmetic. ◦ How can one compute in G ? ◦ For that matter, how can one specify elements of G ? ◦ How can one multiply them? Find inverses? There are two general methods to deal with these questions. (1) The clas- sical approach is to find a good embedding of G into somemain GL n , so every element of G is represented by a matrix. (2) Chevalley’s approach is to represent elements of G more directly in terms of the root datum that defines it. From the root datum one can define Borel subgroups B = TU and the normalizer N G ( T ) . Given representives ˙ w of elements of W = N G ( T ) /T , every element of G can be factored as b ˙ wu in B ˙ wU . With a suitable restriction on u this becomes a unique expression with which one can work. 9/38
The first method works well for the so-called classical groups, which are characterized very nicely in terms of matrices. It has been used also for exceptional groups, although awkwardly. Finding products and inverses is easy, but relating the answer to the structure of G is not simple. In this talk I’ll discuss the second method, in particular a recent contribu- tion due to Robert Kottwitz, building on an old idea due to Jacques Tits, and implemented in programs by myself. This method is closely related to the structure of G , but it is not so easy to perform group operations. 10/38
In truth, I have already deceived you. It turns out that computing in the group reduces quickly to computing in its Lie algebra. (This was already known to Chevalley.) I shall probably say little about the group, and a lot about the Lie algebra. It is in the Lie algebra that the principal and inter- esting difficulties arise. 11/38
The most interesting part of the subject originates with Chevalley. Suppose g to be a semi-simple Lie algebra over C . Chevalley explained how to assign a Z -structure to g , one which led (famously) in turn to his construction of split groups over arbitrary fields. epinglage) for g , which is to say a triple ( b , t , { e α } α ∈ ∆ ) . Choose a frame (´ All choices are equivalent. For each α there exists a good copy of sl 2 , mapping � � � � � � 0 1 0 0 1 0 �→ e α , �→ e − α , �→ h α . 0 0 − 1 0 0 − 1 There exists a unique involution θ acting as − I on t and mapping each e α to e − α . For classical Lie algebras this takes x to − t x . One can find elements e λ in g λ such that e θ λ = e − λ . These are unique up to sign. They make up part of an invariant integral basis. 12/38
If λ , µ , λ + µ are all roots, then [ e λ , e µ ] = N λ,µ e λ + µ for some structure constant N λ,µ . Theorem. (Chevalley) For an invariant integral basis N λ,µ = ± ( p λ,µ + 1) . Here p λ,µ is the string constant: p λ,µ = 0 p λ,µ = 0 1 µ µ λ λ p λ,µ = 0 1 2 p λ,µ = 0 1 2 3 µ µ λ λ 13/38
Chevalley’s theorem is (literally) wonderful. Corresponding to λ is an em- bedding of Q into G : t �− → exp( te λ ) . Chevalley’s theorem implies that t m · N λ,µ N λ,λ + µ . . . N λ, ( m − 1) λ + µ � exp(ad te λ ) e µ = · e mλ + µ m ! m ≥ 0 makes sense. It is a finite series, and equal to exp( te λ ) e µ exp( − te λ ) . With a little work, this makes the string lattice spanned by the e mλ + µ into a representation of SL 2 ( Z ) , and eventually allows you to define G ( Z ) . 14/38
Already, I can bring up my first mystery. Chevalley’s results eventually lead to a construction of a smooth group scheme defined over Z asso- ciated to every root datum. This was begun by Chevalley himself, elab- orated by Demazure and Grothendieck in SGA 3, and has most recently been redone in some very thorough lecture notes by Brian Conrad (based on his lectures at a summer school in Luminy, 2011 that was devoted to updating SGA 3). One part of this business is existence—constructing a split reductive group scheme over Z corresponding to a given root datum. At the AMS Boulder conference of 1965, Kostant proposed an elegant and direct construction of the affine ring of this scheme. Unfortunately, he did not give details. In his Yale lecture notes Robert Steinberg appar- ently used Kostant’s ideas in order to construct the group G as an alge- braic group. (Curiously, as far as I can see, neither Kostant nor Steinberg refers to the other.) I find Steinberg’s discussion unsatisfying, and Conrad doesn’t say much about it. Is Kostant’s construction correct? Conrad tells me this is highly unlikely, but I don’t think anybody really knows one way or the other. 15/38
I can even bring up my second mystery, although it lies further afield. The involution θ determines a Z -structure on G , but also a maximal compact subgroup K of G ( R ) . In some sense, the groups K and Γ = G ( Z ) are hence tied together. (This is a fact that is transparently true for p -adic groups.) There should be a strong version of this assertion in terms of Arthur’s partition of Γ \ G/K . There are questions, too, about the fine structure of this partition when G is not split. Some relationship with ramification and discriminants, strongly suggested by work of Ulrich Stuhler and Dan Grayson on arith- metic groups and stability of lattices. 16/38
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