2-Tori in E8

Mathematische Annalen

Math. Ann. 278, 29 39 (1987)

@ Sprmger-VerlagI987

2-Tori in E 8 J. F. Adams Department of Pure Mathematics and Mathematical Statistics, University, Cambridge CB2 1SB, UK

Dedicated to Friedrich Hirzebruch 1. Introduction A p-torus in a compact Lie group G is a subgroup A C G which is an elementary abelian p-group, A_~Z r x Zp x ... x Zp. M a n y authors, from Borel [1] to Quillen [2], have shown that they influence the behaviour of H*(BG; Fp); this is only one of the reasons for studying them. In this note I will show that one can handle the worst case. Theorem 1.1. Maximal 2-tori in the compact exceptional group E 8 fall into just two conjugacy classes, namely the classes of the examples O(T 8) ( o f rank 9) and EC 8 (of rank 8) which will be given explicitly in Sect. 2. The proof does not involve any preliminary study of ET; on the contrary, (1.I) enables one to study 2-tori in E 7 afterwards, by remarking that every 2-torus in E 7 is a 2-torus in Es. The p r o o f does rely on the fact that E s has a subgroup of local type D 8. Here we recall that the centre of Spin(l 6) is Z 2 x Z 2. Therefore Spin(| 6) has three quotients S p i n ( 1 6 ) / Z 2 : the obvious quotient SO(16)= Spin(l 6)/{1, z}, and the two semi-spin groups Ss +(16) = Spin(16)/{ 1, z + },

Ss (16)=Spin(16)/{l,z

}.

The last two are isomorphic (under the outer a u t o m o r p h i s m of Spin(16)); but it is usual to choose the m a p S p i n ( 1 6 ) ~ E 8 so that its image is Ss+(16). In Sect. 3 we will determine the maximal 2-tori in Ss+(n), after a preliminary study of the projective groups PO(n) and PSO(n). Since we need only the case n = 16, we assume for convenience n - 0 mod 8 and n > 8; this excludes the cases n = 4, 8 which are special, and the case n -= 4 m o d 8, n > 4 which is of less interest. The answer for Ss+(n) is given modulo the answer for Spin(n) a problem usually regarded as a nuisance which grows exponentially with n. However, we need only the case n = 1 6 : this is treated in Sect. 4, and the result for E 8 follows in Sects. 5.

30

J.F. Adams

2. Construction of Examples A n y c o m p a c t connected Lie g r o u p G contains a maximal torus T. The solutions of t 2 = 1 in T form a 2-torus A ( T ) of the same rank as T, say I. O f course, it is not necessarily maximal. Suppose n o w that the Weyl g r o u p W = N ( T ) / T contains the element - 1 ; that is, there exists x ~ N ( T ) such that x t x - 1 = t - ~ for all t e T. T h e n x c o m m u t e s with A(T). We p r o p o s e to form the s u b g r o u p generated by x and A(T), (cf. [3, p. 139]); but before doing so, we need a few remarks a b o u t x. The element x is unique up to conjugacy in G fixing T. F o r any other choice has the form xu for some u e T, and in T we can solve v 2 = u, whence vxv ~= xv 2 =XU.

The element x has order 4 at most. In fact x Z t x - 2 = t, SO X2 ~ T; thus x- x 2 9x = x - 2 , and x 4 = 1 . The square x 2 is independent of the choice of x and lies in the centre of G. F o r the first point, ( x u ) ( x u ) = x Z u lu = x . F o r the second, it follows that x 2 is invariant u n d e r all a u t o m o r p h i s m s of G preserving T, in particular, u n d e r W; hence x 2 lies on every plane of the diagram. It is certainly possible for x to have order 4 [as in the case G = Sp(n)]. However, we assume that x has order 2. F o r example, it is so in the following cases. (a) SO(4m + c) and Spin(8m + e) where c = - l, 0, 1. (b) G2, F4, A d E 7, E s. In such cases, x generates with A ( T ) a 2-torus of rank l + 1, which we call D(T) (for " d o u b l e 2-torus"). If we need to display I or G we write D ( T t) or D(T)C G. This explains the n o t a t i o n D(T s) in (1.1). In particular, the construction gives a 2-torus D(T4)cSpin(8). The m a p s Spin(8) x S p i n ( 8 ) ~ Spin(16) Spin(8) x Spin(8)--*Ss+(16) C E8 have kernels Z 2 and Z 2 x Z 2 respectively; these kernels lie in the centre of Spin(8) x Spin(8), so they lie in D ( T 4) x O(T4). T h u s we obtain the following quotients of D(T4) x D ( T 4 ) . First, a 2-torus of r a n k 9 , E c g c S p i n ( 1 6 ) . Secondly, a 2-torus of rank 8, E C s C Ss + (16) C Es. This explains the n o t a t i o n E C 8 in (1.1). The following c o n s t r u c t i o n is less convenient for o u r proof, but does m o r e to justify the letters E C for "exotic candidate". We have a 2-torus of rank 8 D ( T 4) x D ( T Z ) c F4 x G2C Es; this is conjugate to E C s, either by direct inspection, or by (1.1) [-since the m e t h o d s of Sect. 5 s h o w that it is not subconjugate to D ( T S ) c E s ] .

3. 2-Tori in

PO(n), PSO(n), and Ss+(n)

First we m u s t explain the invariants we use in describing these 2-tori. Typically we have to consider a 2-torus A in a c o m p a c t Lie g r o u p G which comes with a d o u b l e cover Z 2 --~ G --~ G .

2-Tori in Es

31

Then by pullback we get an extension Z 2---~ A---~ A '

and this is classified by an element qe HZ(BA; Z2); we write it q because we often consider it as a quadratic form on A. In particular, we have to consider the following three extensions Zz--~G---*G.

Z 2 ~ O(n) ~ PO(n)

(neven)

Z2~Ss+(n)-~PSO(n) } Z2~Ss-(n)~PSO(n )

(n-=0m~

We write v, v +, v for the corresponding quadratic forms q on ACG. We note that i r a lies in PSO(n), then the quadratic form v [to which it is entitled as a subtorus of PO(n)] becomes v + + v-.

Proposition 3.1. Assume n even. Then maximal 2-tori in PO(n) = O(n)/{ + I} fall into conjugacy classes corresponding to the solutions (r, s) of r. 2s= n with r ~=2. The 2-torus A(r, s) corresponding to (r, s) is q['rank r - 1 + 2s; its quadratic form v is of rank 2s and plus type, that is, equivalent to X l X 2 ~-X3X4 J - . . . - ~ - X 2 s

1X2s 9

Of course, O(n) has a maximal 2-torus unique up to conjugacy, namely the subgroup A of diagonal matrices with diagonal entries + 1. Its image in PO(n), that is A / { + I } , appears in (3.1) as the unique case with v=O, that is A(n,O). We turn to PSO(n), and here we expect to find SA/{ ++_I}, where SA =Ac~SO(n). Proposition 3.2. Assume n = O m o d 8 and n> 8. Then maximal 2-tori in PSO(n) fall into the following coniugacy classes. (Case s=O). The class of SA/{ +_I}. (Case s= 1). The class of D(T)C PSO(n) (see Sect. 2). ( General case). For each solution (r, s) of r. 2~= n with r =I=2 and s > 2, two classes conjugate in PO(n) to A(r,s). Of these, A(r,s) + has c+/~:= 0

of rank 2sandplus type,

v =0 v+

of rank 2s and plus type.

while A(r, s)- has

We turn to Ss+(n). Proposition 3.3. Assume n - : 0 m o d 8 and n> 8. Then maximal 2-tori in Ss+(n) fall into the following conjugacy classes. (General case). For each solution (r, s) of r . 2~= n with r ~ 2 and s > 2, the class of A(r,s) +, the counterimage in Ss +(n) of the 2-torus A(r,s) + in (3.2). (Case s = 1). The class of D(T)C Ss+(n) (see Sect. 2). (Case s = 0). The image in Ss +(n) of each conjugacy class of maximal 2-tori in Spin(n) other than D(T) C Spin(n).

32

J.F. Adams

Proof of (3.1). Suppose given a 2-torus A C PO(n). By pullback from Z 2 ~- { +

I} ~O(n)-~PO(n)

we get an extension Z z - + 7t--+ A .

The classification of such "special 2-groups" is known, and is of course the same as the classification of the quadratic forms q to which they correspond. The classification shows that ~] is a central product F 1 * F 2 * . . . * F ,. Here the "central product" G* H involves amalgamating the given central subgroups Z 2 in G and H; each factor F i may be one of the extensions Z2---+Z 2 x Z2-,-+ Z2---+

68

Z 2

--+Z 2 x Z 2

where ,58 is the ordinary dihedral group of order 8; and we are allowed one further factor, which may be either Z2--+ Z 4 -+ or

Z 2

22--+Q8--+Z2 x Z 2

where Qs is the ordinary quaternion group of order 8. It is easy to determine the real representations of such a central product in which the given central subgroup Z 2 acts as {_+1}. This rests on the following considerations. (i) A representation of G * H gives a representation of G x H. (ii) The (complex) representation-theory of G x H is known in terms of the (complex) representation-theory of G and H. (iii) The representation-theory of 68, Z4, and Q8 is both easy and favourable. The outcome is as follows. We write A as G * H, where G is the central product of all the trivial extensions Z2---~ Z 2 )< Z2--+ Z 2

and H is the central product of the remaining, non-trivial extensions. Then any real representation of A, in which the given central subgroup Z 2 acts as { • I }, may be written as a tensor product U(g)~ V, in which U, V are real representations of G, fl in which Z2 acts as { + 1 }, and V is the unique such representation of H which is irreducible over IR. This Vmay be constructed as the following tensor product over IR. For each dihedral factor `58 we take one copy of the usual real 2-dimensional representation of 68, i.e. the representation by the matrices

If there is a copy of Z 4 we take one copy of the usual action of Z 4 on ~ ; alternatively, if there is a copy of Q8 we take one copy of the usual action of Q8 on ~-I.

2-Tori in E 8

33

In particular, all this applies to o u r assumed e m b e d d i n g A C O(n). Indeed, w h a t we have d o n e describes all the 2-tori A in PO(n), up to conjugacy. It remains to determine which of them are maximal. Since G is a 2-torus we can split U into real eigenspaces for the action of G; we can choose a base in U a d a p t e d to this decomposition, and then enlarge G to the 2-torus G' of diagonal matrices with respect to this base. We still obtain a faithful -,. representation of our enlarged g r o u p A , so if A was originally maximal, we m u s t have G = G'. I f H contains a factor Z 4 acting on G, then we can replace it by 68 acting on its usual 2-dimensional representation, and so obtain a faithful representation of a larger g r o u p / ] ' . So this case is excluded. Similarly, if H contains a factor Q8 acting on ~ , we can replace it by Q8 * Q8 acting on IEI from left and right; of course Q8 * Q8 ~ 88 * 88, and this c o n s t r u c t i o n has to give the tensor p r o d u c t of 2 copies of the usual 2-dimensional representation of 68. Anyway, this case is also excluded. This leads to the description of 3(r, s). Here our g r o u p ,4 comes as a quotient of (Z2) r x (6s) ~, with (Z2) r acting on U = N J as the g r o u p of diagonal matrices, and s

(68) s acting on V = @11t 2 by s copies of the usual 2-dimensional representation. I

The case r = 2 can be excluded as follows. If G = (Z2) 2 acting on lR 2 as

[0 +1

_+01] '

then we can replace it by a c o p y of 8 8 acting as usual, and so e n l a r g e / i . It remains to show that A = A(r, s) c a n n o t be subjugate to A ' = A(r', s') unless r = r', s = s'. F o r this we consider the quadratic form v. O n A it has rank 2s and isotropic subspaces of dimension r + s - I, and similarly for A'. We can have A C A' only i f s =%s , a n d r + s = r% , + s ., S i n c e r . 2 " = n = r ' . 2 ~*', this leads to / ( 2 ....

1)<s'-s.

This is possible only for s ' - s = 1, r' = t, leading to the casc r = 2 which has been excluded. This proves (3.1). The p r o o f of (3.2) needs some preparation. First, the task of describing v + and v is not as bad as one might expect from the p r o b l e m of simultaneously reducing two general quadratic forms o v e r F 2. L e m m a 3.4. Let A be a 2-torus in PSO(n), where n - 0 m o d 8. Then either (i) r + = 0, or (ii) v

=0,

or

(iii) v + = v - ,

v +=a(a+l),

or (iv)

v =b(b+l),

v ++v

=c(c+l)

for some a, b, c, I o f degree i in H * ( B A ; F2). The reason is that o u r quadratic forms are not general; they come from special elements of Hz(BPSO(n); F2). In fact, from this point on our extensions Z2~C,~G

34

J.F. Adams

all have G connected. In this case the covering (~-~G corresponds to a h o m o m o r p h i s m ~I(G)--*Z2, or equivalently ~2(BG)-~Z2, yielding a unique element qeH2(BG; Z2). On any 2-torus ACG this restricts to the element q ~ HZ(BA 9 Z2) we considered before, we omit the proof, which is easy. All this parallels what one usually does for the covering

Z 2~Spin(n)--*SO(n) and the characteristic class w 2. In fact, the characteristic classes v +, v-eHZ(BPSO(n); Z2) both map to w 2~H2(BSO(n); Z2); and if we omit the symbols for the induced maps, the characteristic classes for the extensions Z z --, Spin(n)--* Ss +(n) Z 2~ Spin(n)--* Ss - (n) arev=v

a n d v = v +.

Lemma 3.5. The classes v +, v

in HZ(BPSO(n); Z2) sati~[y

v + ' S q ' v - + v "Sqlv+=O Sq2Sqlv+ + v + . S q % + + S q 2 S q l v - + v - . S q l v - = O

if if

n-0mod8 n~4mod8.

This pinpoints a crucial difference between the cases n - 0, 4 mod 8.

Proof of (3.5). First we dismiss the case n = 4 , in which we have PS0(4)~-S0(3) x SO(3). Then H*(BPSO(4); Z2) is a polynomial algebra on generators v +, Sq I v +, v-, Sq~v -, and the relations Sq2SqXv + +v + .Sqlv + = 0 Sq2Sqlv

+v

"Sql~) = 0

hold separately. Consider now the fibering F = B Spin(n)--* E = B P S O ( n ) ~ B . The base B is an Eilenberg-MacLane space of type (Z 2 x Z 2, 2), with fundamental classes v +, v- ~ HZ(B; Z2). We assume n > 4, so the fibre F has a fundamental class f ~ H 4 ( F ; Z2). A priori we have

zf=~+SqZSqlv + +~q+v+.Sq% + + 7 + v + "Sqlv -

+o~-SqZSqlv +fl v - . S q l v + 7 - v - "Sqlv +

for some coefficients c~+ .... ,7 which remain to be determined. By symmetry under the outer automorphism of PSO(n) we have ~+=c~

,

/~+=/~-,

~+=~

~+=/~+,

~ =I~-,

~+=?'

Since Sq ~j"= 0 we have

2-Tori in Es

35

Thus

Tf =~(Sq2Sq'v + \ + v +'Sqlv + +7(v + . S q l v

-t-Sq2Sq'v-~ +v " S q l v J +v "Sqlv+).

Next we consider the m a p S0(4) • , S0(4) x . . . x S0(4) |

SO(4n).

This passes to the quotient to give

PSO(4) c~ ~PSO(4n). F o r n odd, (BO)* carries v + to v +, v to v - ; this gives 7 = 0. F o r n even, (BO)* carries v+ to0, v toy ++v ;thisgivesc~=0. Finally we consider the fibering

F = B Z z ~ E = BSO(n)--*B = BPSO(n). I claim that w 4 ~ H4(E; Z2) c a n n o t c o m e from H4(B; Z2). In fact, if n -= 4 m o d 8 then the restriction o f w 4 to F is non-zero; thus w 4 is of filtration 0. This does not h a p p e n if n - 0 m o d 8 ; but then w 4 restricts to zero on the Z 2 s u b g r o u p generated by ( - I 2 ) @ I n 2, and restricts n o n - z e r o on the Z 2 s u b g r o u p generated by 1 2 @ ( - I , 2) a l t h o u g h these s u b g r o u p s b e c o m e the same in PSO(n). So in this case w 4 is of filtration 2. If we had c~=0 and ~,=0 then this spectral sequence w o u l d prove H4(B; Z2) ~ H 4 ( E ; Z2) epi. This is a c o n t r a d i c t i o n ; so e = 1 or 7 = 1 according to the case. This proves (3.5).

Proof of (3.4). By (3.5), the e q u a t i o n v +'Sqtv

=v-'Sqlv +

holds in H*(BA; Z2), which is a p o l y n o m i a l algebra and therefore a unique factorisation domain. This leads to the following possibilities only. (i) v + = 0 . (ii) v

=0.

(iii) v + = v . (iv) There is an element l of degree 1 in H*(BA; F2) such that

Sqlv +=lv +,

Sqlv

=lv

.

(v) There are elements l, m, n of degree 1 and q of degree 2 such that

v + =lm,

Sqlv - =nq

v =ln,

Sqlv + =mq.

In case (v) we w o r k out that if l, m, n are all n o n - z e r o then

q=l(l+m),

q=l(l+n)

and m = n. So case (v) is covered by cases (i), (ii), (iii).

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J.F. Adams

By running through the classification of quadratic forms q, we see that the equation

Sqlq = Iq can hold only ifq=mn for some elements m, n of degree 1 (which may be equal or zero) and then I = m + n. So case (iv) leads to the result stated in case (iv) of (3.4).

Proof of (3.2) It is clear from (3.1) that any 2-torus in PSO(n) is subconjugate in PO(n) to A(r,s)c~PSO(n) for some (r,s) with r. 2S=n, r:#2. The case s = 0 gives A(n, O)~PSO(n) = SA/{ + I}; for s > 0 we have A(r, s) C PSO(n). We show that SA/{ + I I cannot be subconjugate in PO(n) to any A(r', s') with s ' > 0. For this we repeat the last part of the proof of (3.1). The inequality from the rank of the isotropic subspaces becomes n - 1 __ 4. It remains to see whether the class of A(r, s) under conjugacy in PO(n) falls into one class or two under conjugacy in PSO(n). The subgroup SA/{ +_I} maps to itself under conjugation by any diagonal matrix of determinant - I. F o r A(89 1) = D(T), the notion of a "double 2-torus" D(T) is invariant even under outer a u t o m o r phisms. F o r s=>2 the rank of v=v + +v- is at least 4; L e m m a 3.4 shows that either v + = 0 or v - = 0, and this settles the question.

Proof of (3.3). Any 2-torus in Ss+(n) maps to some 2-torus in PSO(n), so we must consider the possible maximal 2-tori in PSO(n) and look for the maximal isotropic subspaces of v +. (General case). F o r A(r, s) + the maximal isotropic subspace is clearly A(r, s) + itself. For A(r, s) , any subspace on which v + vanishes lifts to a 2-torus in Spin(n), which brings us to the final case "s = 0". (Case s = 1). Half the elements of A(T)CPSO(n) lift to elements of order 2 in Ss +(n), and (since we assume n-= 0 m o d 8) all the elements of D(T)C PSO(n) which are not in A(T) lift to elements of order 2 in Ss§ There are two choices for a maximal isotropic subspace, but either way, the lift is a D(T) in Ss+(n). (Case s =0). The 2-torus SA/{ + I} already lifts to SO(n), so any subspace which lifts to Ss+(n) must lift to Spin(n). In this case it is clearly enough to consider maximal 2-tori in Spin(n). We can exlude D(T)CSpin(n) because it m a p s to a proper subgroup of D(T)CSs+(n). This shows that any maximal 2-torus in Ss+(n)is one of the 2-tori listed in (3.3). It remains to show that the 2-tori listed in (3.3) are not subconjugate to one another. If one A(r,s) + contained another A(r',s') + (up to conjugacy), then the same would hold for their images in PSO(n), contradicting (3.2). Similarly, let write A o for the image in Ss+(n) of a maximal 2-torus in Spin(n) (case s = 0). If one such Ao contained another A'o (up to conjugacy), then the same

2 - T o r i in E 8

37

would hold for their counterimages in Spin(n). Also A o cannot contain D(T) or an A(r,s) +, because v = v - is zero on A o and non-zero on D(T), A(r,s) + Similarly, D(T) cannot contain any A(r, s) + , becuase v = v- has greater rank on the latter. If D(T) contains an A o, then (by counting ranks) A o must be a maximal isotropic subspace of D(T), in which case A 0 comes from a D(T)C Spin(n); and that case is excluded. An fl(r,s) + cannot contain any Ao, because A o has rank 89 and in A(r,s) + the rank of a maximal isotropic subspace is r+s; the inequality 89 cannot be satisfied for n > 8. Afortiori, an A(r, s) + cannot contain D(T), because that contains an Ao of the sort just excluded. This proves (3.3).

4. 2-Tori in Spin(16) and Ss+(16) Proposition 4.1. Maximal 2-tori in Spin(16)Jdll into just two con]ugacy classes, namely the classes of the examples D(TS)cSpin(16) and EC 9 described in Sect. 2. Corollary 4.2. Maximal 2-tori in Ss+(16)fall into the following four eonjugacy classes. (0) EC ~ of rank 8 (see Sect. 2). (i) D(T) of rank 9. (ii) /](4, 2) + of rank 8. (iv) 4(1,4) + of rank9. This will follow immediately from (3.3), (4.1).

Proof of (4.1). First we show that the two examples are not conjugate. Let "class 2r" be the conjugacy class in Spin(n) of elements whose images in

SO(n) have 2r eigenvalues - 1 and n - 2r eigenvalues + 1 (where 0 < r < 2n). Then the 256 elements which are in D(T s) C Spin(16) but not in A(T 8) are all of class 8. So all the elements of class 4 lie in a proper subgroup, namely A(T). By contrast, in D(T) C Spin(8) the 28 elements of class 4 generate the remaining 4 elements. It follows that in EC9cSpin(16), the elements of class 4 generate the whole group. We now start the main proof. Any 2-torus A in Spin(16) must project to some 2-torus A in SO(16), and after conjugation we may suppose ACSA. The first problem is then to locate the maximal isotropic subspaces of w2, considered as a quadratic form on the vector-space SA of dimension 15. F o r this we need to classify wz according to the theory for classifying quadratic forms over F2; the answer is that w2 is of rank 14 and plus type, equivalent to X 2 X 3 q - X 4 X 5 -~- " " -{- X 14X 1 5

(independent of x 1). Thus the maximal isotropic subspaces are all of rank 8; so the maximal 2-tori in Spin(I 6) must all be of rank 9, and the 2-tori of rank 9 named in (4.1) must be maximal.

38

J.F. Adams The number of maximal isotropic subspaces is (2 ~

1) (21 +1) (22+ 1) (23+ 1) (24+1) (25+1) (26+ 1)

=2.3-5.9-17-33.65 = 2 - 3 4 . 52. 11 913.17. We wish to classify them under conjugacy in SO(16), or equivalently, under the action of the symmetric group 1;~6 permuting the coordinates of R ~6. With sufficient care one can determine the stabiliser in Z16 of our two examples. The number of isotropic subspaces in the Z16-orbit of D(T) turns out to be 16! -2.34.52.7.11.13. 8!27 F o r EC 9 we get

16! = 2 2 . 34. 53. 11 913. 2(14.12.8) 2 Since 7 + 2-5 = 17, the total for these two orbits is 2 . 3 4 . 52. 11 913.17; that is, any maximal isotropic subspace belongs to one of these two orbits. This proves (4.1). 5. 2-Tori in E 8

Lemma 5.1. Any 2-torus in E 8 is conjugate to one in Ss+(16).

Proof First suppose that a 2-torus A acts on a compact Lie group G. If T is maximal among tori preserved by A, then N T / T i s 0-dimensional; in fact, the usual proof for A = 1 carries over, because if N T / T is positive-dimensional, the tangent vector which one uses at T / T may as well be an eigenvector for A. It follows that such a T is maximal in the usual sense; this shows that A preserves at least one maximal torus T in G. (This gives a simple proof for 2-tori of a result true for more general A [3].) It follows that ifA is a 2-torus in G, then A is conjugate to a subgroup of N(T). Then the image o f / t of A in W = N(T)/T is again a 2-torus, and after conjugation we can suppose that A lies in our favourite Sylow 2-subgroup of W. For example, our favourite Sylow 2-subgroup of W(E8) is a Sylow 2-subgroup of W(Ss+(16)). This conjugates A into the normaliser N(T) in Ss+(16). Lemma 5.2. Any 2-torus maximal in Ss+(16) remains maximal in E s.

Proof Such a 2-torus must contain the non-trivial central element z of Ss+(16); so any larger 2-torus must lie in the centraliser of z, i.e. in Ss+(16). To prove (1.1), it remains only to show that of the 2-tori listed in (4.2), the two of rank 9 become conjugate in E 8, and so do the two of rank 8.

2-Tori in E8

39

F o r this, we need to know that elements of order 2 in E 8 fall into just two conjugacy classes. Elements of class "a" have centraliser of local type A 1 • E 7, and their shortest representatives in L(T) have n o r m -~. F o r example, "class 4" in Spin (16) is represented by (89 89 0, 0, 0, 0, 0, 0) with respect to the usual coordinates in L(T), and maps to class "a". Elements of class "b" have centraliser Ss+(16), and their shortest representatives in L(T) have n o r m 1. For example, "class 8" in Spin(16) is represented by (89 89 89 89 0, 0, 0, 0), and maps to class "b". Again, the nontrivial element z in the centre of Ss+(l 6) maps to class "b", either because it is represented by (1, 0, 0, 0, 0, 0, 0, 0) or because its centraliser is Ss+(16). Next let A be one of the 2-tori D(T), EC 8 in Ss+(16). I will show that in either case, the non-trivial element z in the centre of Ss+(16) lies in a proper subgroup B C A which can be distinguished even in E 8. In D(T)CSs+(16), z lies in A(T), which can be distinguished even in E~ as the subgroup generated by the 120 elements of class "a" in D(T). [The elements in D(T) which are not in A(T) are all of class "b".] The case of EC 8 is a little more subtle. In E 8 a 2-torus of rank 3, which consists of the identity and 7 elements of class "a", can be distinguished from one which contains at least one element of class "b". We call the former "2-tori of type G2" , thinking of the example D(T 2) C G 2 C E 8. Both the original copies of D(T4)C Spin(8) inject into EC 8. In EC 8 we have 56 elements of class "a", 28 in one copy of D(T 4) and 28 in the other; let x be one of them. Of the remaining 55 elements of class "a" in EC ~, we can distinguish between the 24 which lie with x in some subgroup of type G2, and the 31 which do not. The former 24 generate the copy of D(T 4) containing x. So we can distinguish a pair of rank-5 subspaces, namely the two copies of D(T4). So we can distinguish their intersection, that is, the c o m m o n image of the centres of the two copies of Spin(8). This gives a distinguishable rank-2 subspace B containing z. Given the subgroup B, we choose an element x ~ A of class "b" which is not in B. (There are 256 choices for x when A = DT, 196 when A = ECS.) In E s, x is conjugate to z; choose a conjugation c~throwing x on z. Since A centralises x, c~A centralises z, so it lies in Ss+(16). Thus c~A is a maximal 2-torus in Ss +(16), of the same rank as A, and the element z does not lie in its distinguished subgroup ~(B). By (4.2), ~A is of type ,4(1,4) + or 4(4,2) + according to its rank. This completes the proof of (1.1).

References 1. Borel, A.: Sous-groupes commutatifs et torsion des groupcs de Lie compacts conncxes. Tokoku Math. J. 13, 216 240 (1961) 2. Quillen, D.: The spectrum of an equivariant cohomology ring. I, I!. Ann. Math. 94, 549 572, 573 602 (1971) 3. Borcl, A., Serrc, J.-P.: Sur certains sous-groupes des groupes de Lie compacts. Comment. Math. Helv. 27, 128 139 (1953) Receivcd July 26, 1986