Reflection Groups on the Octave Hyperbolic Plane

Journal of Algebra 213, 467᎐498 Ž1998. Article ID jabr.1998.7671, available online at http:rrwww.idealibrary.com on

Reflection Groups on the Octave Hyperbolic Plane Daniel AllcockU Department of Mathematics, Har¨ ard Uni¨ ersity, Cambridge, Massachusetts 02138 E-mail: [email protected] Communicated by Efim Zelmano¨ Received August 12, 1997

For two different integral forms K of the exceptional Jordan algebra we show that Aut K is generated by octave reflections. These provide ‘‘geometric’’ examples of discrete reflection groups acting with finite covolume on the octave Žor Cayley. hyperbolic plane ⺟H 2 , the exceptional rank one symmetric space. ŽThe isometry group of the plane is the exceptional Lie group F4Žy 20. .. Our groups are defined in terms of Coxeter’s discrete subring K of the nonassociative division algebra ⺟ and we interpret them as the symmetry groups of ‘‘Lorentzian lattices’’ over K. We also show that the reflection group of the ‘‘hyperbolic cell’’ over K is the rotation subgroup of a particular real reflection group acting on H 8 ( ⺟H 1. Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix. 䊚 1999 Academic Press

1. INTRODUCTION The octave hyperbolic plane ⺟H 2 is the exceptional rank one symmetric space; it is very similar to the more familiar real Žand complex and quaternionic. hyperbolic spaces. There is a natural notion of a hyerplane in ⺟H 2 , and of a reflection in such a hyperplane Žthe mirror of the reflection .. In this paper we explain this and construct two discrete groups of isometries of ⺟H 2 that are generated by reflections. One can also define the ‘‘octave hyperbolic line’’ ⺟H 1, and it turns out to be isometric with the real hyperbolic space H 8. In this setting, a hyperplane is just a point and the octave reflection therein is just central inversion. We will construct a discrete group of isometries of ⺟H 1 generated by octave reflections and show that it is the rotation subgroup of a certain group of isometries of H 8 generated by real reflections. This real reflection group is easy to describe; it has a simplex for its fundamental domain, with U

Supported by an NSF postdoctoral fellowship. 467 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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DANIEL ALLCOCK

Coxeter diagram 䢇

䢇

䢇

䢇

䢇

䢇

䢇

䢇

䢇

We construct all three of our groups by defining them arithmetically and then finding sufficiently many reflections to generate them. It turns out that in each case one can easily write down a large set of reflections preserving the relevant algebraic structure, and that the mirrors of these reflections are arranged in a pattern governed by the combinatorics of the E7 and E8 root lattices. We use facts about the geometry of these lattices to prove that these ‘‘known’’ reflections actually generate the entire group. To the author’s knowledge this is the first geometric construction of discrete groups of isometries of ⺟H 2 . To discuss ⺟H 2 in any depth one must introduce various spaces of 3 = 3 matrices with entries in the nonassociative field ⺟ of octaves, such as the space JI of Hermitian matrices; such matrices should informally be regarded as ‘‘Hermitian forms’’ on the nonexistent vector space ⺟ 3. One can define the determinant of such a ‘‘Hermitian form’’ in terms of a certain cubic form. The group HIK of linear transformations of JI that preserve the determinant form is isomorphic to the exceptional Lie group E6Žy 26. . This representation of HI is very closely analogous to the action of PGL3 ⺓ on the space of 3 = 3 complex Hermitian matrices given for g g GL3 ⺓ by ⌽ ¬ g ⌽ g U . ŽOne can replace ⺓ here with any associative ring.. One should think of HI as ‘‘PGL3 ⺟,’’ and the stabilizer in HI of some matrix ⌽ as the unitary group preserving the ‘‘Hermitian form on ⺟ 3 ’’ with inner product matrix ⌽. For appropriate choice of ⌽ this stabilizer is isomorphic to G s F4Žy 20. s Aut ⺟H 2 , and one can use this to construct ⺟H 2 . To construct discrete subgroups of G one can simply take ⌽ to be integral and consider the stabilizer of ⌽ in the subgroup HIK of HI that preserves the set of integral Hermitian matrices. By an integral matrix we mean one whose entries lie the natural discrete subring K of ⺟ discovered by Coxeter w9x. In light of the above interpretation of G and HI , we may think of this discrete subgroup as being essentially the isometry group of the ‘‘lattice’’ over K with inner product matrix ⌽. Our main result is that if ⌽ is

ž

1 0 0

0 0 1

0 1 0

/

or

ž

1 0 0

0 0 2

0 2 0

/

Ž 1.1.

then the discrete group is generated by octave reflections. We note that the first of these groups has been studied by Gross w13x in a different guise Žsee Section 7.. If one uses 2 = 2 matrices over ⺟ then one finds that the

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analogous discrete subgroup defined by the matrix 01 10 is also generated by octave reflections, and that it has index 2 in the real reflection group with Coxeter diagram given above. We note in passing that the finite simple group 3 D4 Ž2. described in the ATLAS w7x as an octave reflection group in the compact form of F4 has been realized by Elkies and Gross w10x as the stabilizer in HIK of a certain positive definite matrix with entries in K and unit determinant. Our construction of reflection groups in F4Žy 20. reveals the source of the ideas for this investigation. In w1x we constructed a large number of discrete reflection groups acting with finite covolume on complex and quaternionic hyperbolic spaces ⺓ H n and ⺘ H n. These groups were defined as the automorphism groups of certain Lorentzian lattices over discrete subrings of ⺓ and ⺘. ŽA Lorentzian lattice is a free module equipped with a Hermitian form of signature yqq ⭈⭈⭈ q. See w1x for details.. This work was in turn inspired by the work of Vinberg w18, 19x, Vinberg and Kaplinskaja w20x, Conway w6x, and Borcherds w4x on real hyperbolic reflection groups defined in terms of Lorentzian lattices over ⺪. The basic idea of w1x was that the symmetry group of a Lorentzian lattice sometimes contains a collection of reflections whose mirrors are arranged in the pattern of some positive-definite lattice. If this lattice is ‘‘good enough’’ Že.g., has small covering radius. then these reflections generate a group with finite covolume. We use the same idea here, although all of the arguments of w1x must be changed because ⺟ and K are not associative and so modules and lattices over them do not make sense. We are pleased that the geometric ideas continue to work in the nonassociative case even though all of the formalism must be rewritten. In this paper, the relevant positive-definite ‘‘lattice’’ over K is just K itself, which is a scaled copy of the E8 root lattice. Above, we described F4Žy 20. as the stabilizer in E6Žy26. of any of various Hermitian matrices. However, for computations it is more convenient to describe the group as the automorphism group of a Jordan algebra J and to describe ⺟H 2 in terms of the idempotent elements of J. For any 3 = 3 nondegenerate real symmetric matrix ⌽ we define J⌽ to be the set of 3 = 3 octave matrices that are Hermitian with respect to ⌽. This vector space is closed under the Jordan multiplication X )Y s Ž XY q YX .r2, and the automorphism group of J⌽ turns out to be a real form of F4 . In particular, if ⌽ is indefinite then the group is F4Žy 20. s Aut ⺟H 2 . For most of the paper we will work with J ⌿ and various integral forms thereof, where

ž /

1 ⌿s 0 0

ž

0 0 1

0 1 . 0

/

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DANIEL ALLCOCK

This is a convenient setting for computations and for the construction of the reflection groups. A little bit more effort is then required to identify the groups as the stabilizers in HIK of the Hermitian forms Ž1.1.. This extra work essentially consists of the introduction of the determinant forms on JI and J ⌿ and an identification of these cubic forms with each other. Existing treatments of ⺟H 2 are either elegant and abstract but not suited for computation Žsee w16x. or else are defined in terms of the Jordan algebra Jdiagwy1, q1, q1x Žsee w15x.. To prove our results we need to introduce and name various transformations of J ⌿ and make other detailed constructions. In light of this, our treatment of J is almost entirely self-contained. In Section 2 we describe ⺟ and list some identities useful for computing therein. We then heuristically describe in Section 3 the geometry of ⺟H 2 , to guide the reader through the algebraic thicket of Section 4, which formally defines the Jordan algebras and ⺟H 2 , provides useful coordinates for them, and establishes key properties of their symmetry groups. Section 5 is the heart of the paper and constructs the advertised reflection groups. The proof that the groups are the entire symmetry groups of two integral forms of J requires the rather involved Theorem 5.3, but we indicate how to obtain the weaker result that they have finite covolume without using this theorem. In Section 6 we describe the determinant form on J, a cubic form which makes the inclusion F4Žy 20. : E6Žy 26. ( Aut ⺟P 2 visible. Finally in Section 7 we use the determinant form to realize our reflection groups as the stabilizers of the integral Hermitian forms Ž1.1.. 2. THE NONASSOCIATIVE FIELD ⺟ The algebra ⺟ of octaves is the algebra over ⺢ with basis e⬁ s 1, e0 , . . . , e6 with e a2 s y1 for a / ⬁, any two elements lying in an associative algebra, and e a e aq1 s e aq3 for each a / ⬁, with indices read modulo 7. The algebra of octaves is noncommutative and nonassociative, but otherwise a division algebra. For computational purposes it is useful to know that for distinct a, b, c / ⬁ we have e a e b s ye b e a and e aŽ e b e c . s "Ž e a e b . e c , with associativity holding just if

 a, b, c 4 s  d, d q 1, d q 3 4

Ž mod 7 . for some d s 0, . . . , 6.

Sometimes it is convenient to write i, j, k, or l for e0 , e1 , e3 , and e2 , respectively. Then i generates a copy of ⺓ and i, j, k generate a copy of the quaternions ⺘. The real part Re x of x s Ý x a e a is x⬁ , and the conjugate x of x is x s yx q 2 Re x; we have Re x s Re x. We say that

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x is imaginary if Re x s 0 and we write Im ⺟ for the set of imaginary octaves. The norm < x < 2 of x is defined to be xx s Ý x a2 , and x is called a unit if it has norm one. The identities xys yx and < xy < 2 s < x < 2 < y < 2 hold universally. Since ⺟ is nonassociative, computations with it are sometimes complicated. Five identities we will use are z Ž xy . z s Ž zx . Ž yz . ,

Ž 2.1.

Re Ž Ž xy . z . s Re Ž x Ž yz . . s Re Ž Ž yz . x .

Ž 2.2.

Ž ␮ x ␮ . Ž ␮ y . s ␮ Ž xy . ,

Ž 2.3.

Ž x ␮ . Ž ␮ y ␮ . s Ž xy . ␮ ,

Ž 2.4.

xy q yx s Ž x ␮ . Ž ␮ y . q Ž y ␮ . Ž ␮ x .

Ž 2.5.

and

which hold for all x, y, z g ⺟ and all imaginary units ␮. We write ReŽ xyz . for any of the three expressions in Ž2.2.. Identities such as these are easily proven if one takes advantage of the automorphism group of ⺟. This group is the compact form of the exceptional Lie group G 2 , and the stabilizer of a subalgebra isomorphic to ⺢, ⺓, or ⺘ acts transitively on the units orthogonal to the algebra. ŽSee w12x for a proof.. For example, we prove Ž2.3.. After applying an automorphism of ⺟ we may take ␮ s i, x s x 0 q ␣ j, and y s y 0 q ␤ l with ␣ , ␤ g ⺢, x 0 g ⺓, and y 0 g ⺘. After expanding both sides of Ž2.3. we find that almost all terms cancel by associativity in ⺘. All that remains is to check

Ž ij ı . Ž il . s i Ž jl . , which one does using the definition of multiplication. Note that Ž2.4. may be obtained from Ž2.3. by taking conjugates. We observe that if A1 , . . . , A n are matrices over ⺟, with all except at most two of them having all real entries, then the product A1 ⭈⭈⭈ A n is independent of the manner in which the terms are grouped by parentheses. The same result holds if all the entries of all the matrices lie in some associative subalgebra of ⺟. Finally, we will use the fact that the group of transformations of ⺟ generated by the left Žor right. multiplications by imaginary units act transitively on the unit sphere S 7 in ⺟. ŽProof: every orbit contains an equator S 6 of S 7 : ⺟, so any two orbits meet..

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3. THE GEOMETRY OF ⺟H 2 This section is not logically necessary for those following it. Its purpose is to tell the reader in advance what some of the answers are, to serve as a guide to the rather heavy algebra of Section 4. We discovered the reflection groups mostly by geometric visualization using the model described here, together with analogies with the constructions of w1x. The geometry of ⺟H 2 and ⺟P 2 can be developed entirely synthetically, along the lines of w16, 17x, but we will use the Jordan algebra approach in order to be able to use a theorem of Borel and Harish-Chandra at a crucial point in the proof of Theorem 5.4. Some of the basic ideas described here are implicit in w14x and they are all developed in w11, 12x. Even though ⺟ is not associative and so modules over it can’t reasonably be defined, one should imagine the fictional vector space ⺟ 3. If it were equipped with a nondegenerate Hermitian form ⌽ then we could consider the set J⌽ of ⺟-linear transformations of ⺟ 3 that were self-adjoint with respect to ⌽. Among these would lie the orthogonal projections onto subspaces of ⺟ 3 ; these would be the identity and zero operators together with two continuous families corresponding to the 1- and 2-dimensional subspaces. In particular, thinking of elements of ⺟ 3 as row vectors with matrices acting on the right we could consider X s ¨ U ¨ where ¨ U would be the ⌽-adjoint of ¨ g ⺟ 3. Up to a multiplicative constant this would be the projection onto the span of ¨ . It would turn out that such transformations could be essentially characterized by the conditions that X 2 s ␭ X and Tr X s ␭ where ␭ g ⺢ would be the Žsquare. norm of ¨ . This would allow one to recover the 1-dimensional subspaces of ⺟ 3 and the norms of vectors in them. Furthermore, if X and Y lay in J⌽ then their Jordan product X )Y s Ž XY q YX .r2 would also lie in J⌽ , so that J⌽ would be a Jordan algebra. Finally, one could recover the notion of orthogonality of subspaces of ⺟ 3 because if X and Y were projections to two subspaces then these subspaces would be orthogonal just if X )Y s 0. The reason one goes to these lengths is that ⺟ 3 is not in any sense a vector space over ⺟, so the above account is purely fictional. However, the Jordan algebra J⌽ does exist Žalthough we will only treat the case of real ⌽ .. This allows us to recover much of the structure that ⺟ 3 would have provided if it existed. In particular, if ⌽ has signature yqq then we can recover the set of ‘‘1-dimensional subspaces of ⺟ 3 on which ⌽ is negative definite.’’ By analogy with hyperbolic geometry over ⺢, ⺓, and ⺘ we define ⺟H 2 to be this set. The group Aut J⌽ turns out to be F4Žy20. , which acts transitively and with compact stabilizer on ⺟H 2 . The geometry of ⺟H 2 is very similar to more conventional hyperbolic geometry. In particular, ⺟H 2 is an open 16-ball and it has a natural boundary ⭸ ⺟H 2 ‘‘at infinity,’’ a sphere S 15 which arises from the nilpo-

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tents of J⌽ Žor, heuristically, from the norm 0 elements of ⺟ 3 .. Furthermore, by considering idempotents of J⌽ one obtains points ‘‘outside’’ ⭸ ⺟H 2 which together with ⺟H 2 and ⭸ ⺟H 2 form the octave projective plane ⺟P 2 . This is essentially the same as the realization of real hyperbolic space H n as an open ball in ⺢ P n, as the image of the elements of ⺢ nq 1 of negative norm with respect to a quadratic form of signature yqq ⭈⭈⭈ q. The only difference is that there is no vector space ⺟ 3 associated with ⺟P 2 . We will show that Aut ⺟H 2 acts transitively on ⭸ ⺟H 2 , and the stabilizer of a null point will be very important in our work. There is an upper-half-space model for ⺟H 2 which is ideal for studying the stabilizer of a null point. In this model, the points of ⺟P 2 Žexcept those on the line at infinity. are described by pairs Ž x, z . with x, z g ⺟. The points of ⺟H 2 are the pairs with Re z ) 0, and the points of ⭸ ⺟H 2 are the pairs with Re z s 0, together with one extra point called ⬁, which lies on the line at infinity. There are ‘‘translations’’ stabilizing ⬁, which have the form

Ž x, z . ¬ Ž x q ␰ , z y Im Ž x ␰ . q ␩ . with ␰ g ⺟ and ␩ g Im ⺟. The translations turn out to form a 15-dimensional Lie group H 15 which Žobviously. acts transitively on ⭸ ⺟H 2 _  ⬁4 . The group of translations is closely analogous to the translations of Euclidean space ⺢ n which act on H nq 1 in the usual upper-half-space model. The only substantial difference from the real case is that H 15 is nonabelian, being a sort of octave version of the Heisenberg group. We mentioned above that one can define the notion of orthogonality of ‘‘subspaces of ⺟ 3 ’’ purely in terms of the Jordan algebra. This leads to the concept of a hyperplane in ⺟H 2 ; usually we will call a hyperplane a line. A reflection is a nontrivial transformation of ⺟H 2 that fixes a line pointwise. We will see that there is a unique reflection across each line; the line is called the mirror of the reflection. The map RX : Ž x, z . ¬ Žyx, z . is an example of a reflection which fixes ⬁ and there is another reflection R which exchanges ⬁ with Ž0, 0.. The coordinate expression for R is complicated, but R is analogous to a real reflection of H n whose mirror appears as a hemisphere resting on ⭸ H n in the upper-half-space model. The two reflection groups we will build will be generated by conjugates of R and RX by translations satisfying appropriate integrality conditions on ␰ and ␩. The mirrors of the conjugates of RX will be arranged in the pattern of a discrete subgroup of H 15, and if this subgroup is ‘‘dense enough’’ then the mirrors will ‘‘cover’’ ⭸ ⺟H 2 _  ⬁4 . This will allow us to prove that the groups have finite covolume in Aut ⺟H 2 .

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DANIEL ALLCOCK

4. JORDAN ALGEBRAS We will use angle-brackets ² : to denote the values of various multilinear forms, and also to denote ‘‘the linear span of’’ or ‘‘the group generated by.’’ One of the latter possibilities applies when there are no vertical bars < between the brackets. In this case, the meaning will be clear from the sort of objects lying between the brackets. If ⌽ is a symmetric matrix in M3 ⺢, then we will regard it informally as a ‘‘Hermitian form on ⺟ 3 ’’; we will restrict our attention to invertible ⌽, corresponding to nondegenerate forms. We write M3 ⺟ for the real vector space of 3 = 3 matrices with entries in ⺟ and denote by X U the conjugate-transpose of a matrix X. The vector space J⌽ s  X g M3 ⺟ N X ⌽ s ⌽ X U 4 is closed under the Jordan multiplication X )Y s Ž XY q YX .r2, and we call it the Jordan algebra associated to ⌽. We say that X, Y g J⌽ Jordan-commute if X )Y s 0. We informally regard elements of J⌽ as ‘‘transformations of ⺟ 3 ’’ that are Hermitian with respect to ⌽. In most treatments Že.g., w11, 12x., ⺟P 2 is described in terms of JI , whose automorphism group is the compact form of F4 . We will study ⌽ / I because we are interested in ⺟H 2 and because it is useful to formulate the theory for more general ⌽ᎏthis clarifies the various roles of the matrices involved. For g g GL3 ⺢ we define the transformation C g of M3 ⺟ given by C g : X ¬ gy1 Xg. A quick computation shows that C g carries J g ⌽ g U to J⌽ and preserves matrix multiplication on M3 ⺟. We write O Ž ⌽ . for the subgroup of GL3 ⺢ whose elements are unitary with respect to ⌽ Žthat is, g g O Ž ⌽ . if g ⌽ g U s ⌽ .. Scalars in O Ž ⌽ . act trivially on J⌽ , so we have POŽ ⌽ . s O Ž ⌽ .r "I 4 : Aut J⌽ . When ⌽ is indefinite, POŽ ⌽ . may be identified with the subgroup OqŽ ⌽ . of O Ž ⌽ . that preserves each of the two null cones defined in ⺢ 3 by ⌽. We will assume this identification henceforth. For any X g M3 ⺟ we define ␹ Ž X . s Re TrŽ X . and the symmetric inner product ² X N Y : s ␹ Ž X )Y .. We call ² X N X : the norm of X. It is easy to see that TrŽ XY . s TrŽ YX . if either X or Y is real, so GL3 ⺢ preserves the trace form on M3 ⺟. Since ² X N Y : is defined in terms of the trace and the multiplication, GL3 ⺢ also preserves norms and inner products. We will see that all of Aut J⌽ preserves the restrictions of these forms to J⌽ , and that the trace of any element of J⌽ is real. We will mainly be concerned with the indefinite form 1 ⌿s 0 0

ž

0 0 1

0 1 0

/

and we write J for J ⌿ . For any ⌽, there exists g g GL3 ⺢ such that g ⌽ g U g  "I, " ⌿4 , so J⌽ is isomorphic to JI or to J ⌿ . An element of J

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has the form X s Ž a, b, c, u, ¨ , w . s

ž

a ¨

w

w u c

¨

b u

/

Ž 4.1.

with a, b, c g ⺢ and u, ¨ , w g ⺟. It is obvious that X has real trace. Since all elements of JI also have real trace Žproof: compute, or see w11x. and the maps C g preserve traces, the trace of any element of any J⌽ is real. Computation reveals ² X N X : s a2 q 2 bc q 2 Re Ž u 2 . q 4 Re Ž ¨ w . , and so by polarization we obtain ² X 1 N X 2 : s a1 a2 q b1 c2 q b 2 c1 q 2 Re Ž u1 u 2 . q 2 Re Ž ¨ 1w 2 q ¨ 2 w 1 . .

Ž 4.2. In order to work with the reflection groups of Section 5 we will need detailed information about J and its automorphism group. We begin by giving the multiplication table for J. For each x g ⺟ we define 1 As 0 0

ž

0 Ux s 0 0

0 x 0

ž

0 0 0

0 0 , 0

0 Bs 0 0

/

0 0 , x

/

ž

Vx s

ž

0 x 0

0 0 0

0 0 0

0 1 , 0

0 Cs 0 0

/

x 0 , 0

/

ž

and

0 0 1

0 Wx s 0 x

ž

0 0 , 0

/

x 0 0

0 0 . 0

/

With respect to this spanning set, the Jordan multiplication is given in Table 4.1. Note that entries denote twice the Jordan product. We write U Žresp. V, W . for the span of the Ux Žresp. Vx , Wx .. We write Im U for the TABLE 4.1 Entries Denote Twice the Jordan Product in J

A B C Ux Vx Wx

A

B

C

Uy

Vy

Wy

2A

0 0

0 U1 0

0 2 ReŽ y . ⭈ B 2 ReŽ y . ⭈ C Ux yqy x

Vy 0 Wy Vx y 2 ReŽ xy . ⭈ B

Wy Vy 0 Wy x 2 ReŽ xy . ⭈ A q Uy x 2 ReŽ xy . ⭈ C

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DANIEL ALLCOCK

span of those Ux with x g Im ⺟. The nilpotent B will be very important in our analysis, and we define the height of X g J to be htŽ X . s ² X N B :. For X given by Ž4.1., we see by Ž4.2. that htŽ X . s c. We now introduce several transformations which will turn out to be automorphisms of J. We already know that OqŽ ⌿ . : Aut J. For t g ⺢ and ␮ any imaginary unit of ⺟ we define the transformations R: Ž a, b, c, u, ¨ , w . ¬ Ž a, c, b, u, yw, y¨ .

Ž 4.3.

S␮ : Ž a, b, c, u, ¨ , w . ¬ Ž a, b, c, ␮ u ␮ , ␮¨ , w␮ .

Ž 4.4.

Tt , 0 s C g t

for g t s



1 t 0

0 1 0

yt yt 2r2 1

0

Ž note

gy1 s gyt . . Ž 4.5. t

We will soon introduce a more convenient notation for certain elements of J, which will greatly simplify these expressions. We write G for the group generated by R, the S␮ , and the Tt, 0 . Later in this section we will see that G s Aut J, that the S␮ generate a group isomorphic to Spin 7 ⺢, and that the Tt, 0 together with their conjugates under this Spin 7 ⺢ generate a 15-dimensional nilpotent Lie group. An important element of G is the map RX : Ž a, b, c, u, ¨ , w . ¬ Ž a, b, c, u, y¨ , yw . ,

Ž 4.6.

which is the square of any S␮ . We will define octave reflections in Section 5 and see that RX is one. LEMMA 4.1. The transformations R, S␮ , and Tt, 0 are automorphisms of J and preser¨ e the trace and norm forms. We ha¨ e ² R, Tt, 0 : s OqŽ ⌿ ., and it follows that OqŽ ⌿ . : G. Proof. We first observe that R and Tt, 0 lie in OqŽ ⌿ .: Tt, 0 obviously does and R s C g for 1 gs 0 0

ž

0 0 y1

0 y1 . 0

/

Verification that each S␮ is an automorphism relies on several pages of computation, using the identities Ž2.1. ᎐ Ž2.5. and Table 4.1. It is obvious that S␮ preserves traces and since it preserves the Jordan multiplication it also preserve norms. Alternately, Section 6 defines the determinant form on J and Theorem 6.2 uses it to prove that S␮ g Aut J. This approach reduces considerably the amount of work required but is much more circuitous.

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We write elements of ⺢ 3 as row vectors; Oq Ž ⌿ . acting on such vectors Žby multiplication on the right. preserves the quadratic form ⌿. The g t are the parabolic transformations stabilizing Ž0, 0, 1. and their conjugates by g are those stabilizing Ž0, 1, 0.. It is obvious that these one-parameter subgroups generate SOqŽ ⌿ . ( SOqŽ 2, 1; ⺢.. Since R g OqŽ ⌿ . y SOqŽ ⌿ . we see that ² R, Tt, 0 : s ² g, g t : s Oq Ž ⌿ .. Working with 3 = 3 matrices is tedious and there is a better notation, which closely resembles the use of ‘‘vectors in ⺟ 3.’’ We write elements of the real vector space ⺟ 3 as row vectors, and define ⺟ 03 s  Ž x, y, z . g ⺟ 3 N x, y, z all lie in some associative algebra 4 3 ⺟ 00 s  Ž x, y, z . g ⺟ 3 N y g ⺢ 4 : ⺟ 03 .

Suppose ⌽ is given. We define the map ␲⌽ : ⺟ 03 ª M3 ⺟ by ␲⌽ Ž ¨ . s ⌽ ¨ U ¨ ; we write ␲ for ␲ ⌿ . One checks immediately that ␲⌽ Ž⺟ 03 . : J⌽ . We say that an element of J⌽ is ‘‘good’’ Žwith respect to ⌽ . if it is nonzero and lies in the image of ␲⌽ . One checks that if Ž x, y, z . g ⺟ 03 and ␣ g ⺟ y  04 such that x, y, z, and ␣ all lie in some associative algebra then ␲⌽ Ž x, y, z . 3 s ␲⌽ Ž ␣ x, ␣ y, ␣ z ., so any good element of J⌽ lies in the image of ⺟ 00 . It 3 3 is easy to see that the span of ␲ Ž⺟ 0 . is all of J. The elements of ⺟ 0 and 3 ⺟ 00 have been dubbed ‘‘restricted homogeneous coordinates’’ by Aslaksen w3x. ŽActually, Aslaksen considered a slightly different version of the special case ⌽ s I. A very similar idea appears in w14x. The relation between these ideas is explained in w2x.. For reference, we record that the ordered pair Ž x, z . of Section 3 represents

␲ Ž x, 1, z y < x < 2r2 . g J. ŽWe will not need this identification.. The following theorem a very useful computational tool. THEOREM 4.2. If ¨ , w g ⺟ 03 and all six entries of ¨ and w lie in some associati¨ e subalgebra of ⺟, then for any ⌽ we ha¨ e

² ␲⌽ Ž ¨ .

␲⌽ Ž w .: s < ¨ ⌽ wU < 2 .

Proof. Let ␣ denote the single entry of ¨ ⌽ wU . We write ¨ s Ž ¨ a ., w s Ž w b . and ⌽ s Ž ␾ ab . for a, b s 1, 2, 3. In the derivation below we have associated terms freely and used identity Ž2.2. and the symmetry and

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reality of ⌽.

² ␲⌽ Ž ¨ .

␲⌽ Ž w .: s Tr Ž ␲⌽ Ž ¨ . )␲⌽ Ž w . . s Tr Ž ⌽ ¨ U ¨ ⌽ wU w q ⌽ wU w⌽ ¨ U ¨ . r2 s Tr Ž ⌽ ¨ U␣ w q ⌽ wU ␣ ¨ . r2 s

Ý Ž ␾ab¨ b ␣ wa q ␾ab w b ␣ ¨ a . r2 a, b

s

Ý Re Ž ␾ab¨ b ␣ wa . a, b

s Re ␣ Ý wa ␾ ab¨ b

ž

a, b

/

s Re Ž ␣␣ . s < ␣ < 2 . For ␭ g ⺢, we say that X g J⌽ is ␭-potent Žor is a ␭-potent. if X ) X s ␭ X. We refer to 1-, 0-, and Žy1.-potents as idempotents, nilpotents, and negpotents, respectively. We say that X is potent if it is 3 ␭-potent for some ␭. One reason we use ⺟ 03 and ⺟ 00 is that the good elements turn out to be the most important elements of J⌽ . In particular, every good element is potent. We will see that the geometry of ⺟H 2 may be described in terms of the good potents of J⌽ . We define the norm < ¨ < 2⌽ of ¨ g ⺟ 03 to be the single entry of ¨ ⌽ ¨ U , which is automatically real. Then we have

␲⌽ Ž ¨ . ␲⌽ Ž ¨ . s ⌽ ¨ U ¨ ⌽ ¨ U ¨ s ⌽ ¨ U < ¨ < 2⌽ ¨ s < ¨ < ⌽2 ␲⌽ Ž ¨ . , so we see that ␲⌽ Ž ¨ . is < ¨ < 2⌽ -potent. ŽWarning: The norm < ¨ < 2⌽ of ¨ g ⺟ 03 is not the same as the norm in J⌽ of ␲⌽ Ž ¨ . ᎏby Theorem 4.2 the latter norm is the square of the former. This is an unavoidable inconvenience.. We also have Tr Ž ␲⌽ Ž ¨ . . s Re Tr Ž ␲⌽ Ž ¨ . . s Re

Ý ␾ab¨ b¨ a s Re Ý ¨ a ␾ab¨ b s < ¨ < 2⌽ . a, b

a, b

Therefore a necessary condition for X g J⌽ to be good is that there be ␭ g ⺢ such that X is ␭-potent and has trace ␭. In Theorem 4.3 we will see that this is nearly a sufficient condition. We can define an action of O Ž ⌽ . on ⺟ 03 that is ␲⌽ -equivariant with its action on J⌽ . Namely, g g O Ž ⌽ . acts by ¨ ¬ ¨ g; observe that this preserves ⺟ 03 . Taking ⌽ s ⌿ we will henceforth regard Tt, 0 as acting on ⺟ 03

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by g t g O Ž ⌿ ., and we observe that this action is ␲-equivariant and 3 3 preserves ⺟ 00 . We also define an action of the S␮ on ⺟ 00 by S␮ : Ž x, y, z . ¬ Ž ␮ x, y, ␮ z ␮ . . With the aid of Ž2.3. and the fact that y g ⺢ one can check that this action and the action of S␮ on J are ␲-equivariant. 3 An important set of transformations of ⺟ 00 are the ‘‘translations’’ T␰ , ␩ : Ž x, y, z . ¬ Ž x q ␰ y, y, z y x ␰ y < ␰ < 2 yr2 q y␩ . with ␰ g ⺟ and ␩ g Im ⺟. Those with ␰ s 0 are called central translations. If ␰ s t g ⺢ and ␩ s 0 then this definition agrees with the action of 3 Tt, 0 on ⺟ 00 defined in the previous paragraph. All of the translations fix Ž0, 0, 1., which is useful because B s ␲ Ž0, 0, 1. is a useful nilpotent of J. Using Ž2.1. and the fact that ␮ s y␮ one may show S␮ (T␰ , ␩ ( S ␮ s T␮␰ , ␮␩␮

Ž 4.7.

T␰ 1 , ␩ 1 (T␰ 2 , ␩ 2 s T␰ 1q ␰ 2 , ␩ 1q ␩ 2qImŽ ␰ 1␰ 2 .

Ž 4.8.

T␰y1 , ␩ s Ty ␰ , y ␩

Ž 4.9.

T␰ 1 , ␩ 1 , T␰ 2 , ␩ 2 s T␰y1 (T␰y1 (T␰ 1 , ␩ 1 (T␰ 2 , ␩ 2 s T0, 2 ImŽ ␰ 1␰ 2 . . Ž 4.10. 1 , ␩1 2 , ␩2 These show that the translations form a 15-dimensional Lie group Žwhich we call H 15 . which is nilpotent of class 2 and normalized by the S␮ . Its derived subgroup coincides with its center and consists of the central H 15 is a sort of octave version of the Heisenberg group. translations ᎏH 3 Because ␲ Ž⺟ 00 . spans J, the ␲-equivariance properties of the Tt, 0 and S␮ show that the action on J of any elements of ²Tt, 0 , S␮ : is completely 3 determined by its action on ⺟ 00 . The relations above show that all of 15 H i ² S␮ : is generated by the S␮ and those Tt, 0 with t g ⺢, so all of H 15 i ² S␮ : maps into G : Aut J. One reason for introducing the restricted homogeneous coordinates is that the expression for the action of T␰ , ␩ on J is horrible for general ␰ and ␩. We now show that the natural 3 map from the group ²Tt, 0 , S␮ : acting on ⺟ 00 to the group ²Tt, 0 , S␮ : : G 3 acting on J is an isomorphism. The action of s g ² S␮ : on ⺟ 00 is deter3 mined by its actions on the first and third coordinates of elements of ⺟ 00 . These actions coincide with those of the image of s in G on the subspaces 3 V and U of J. Thus ² S␮ : : Aut ⺟ 00 acts faithfully on J. If h g H 15 y 1 3 and s g ² S␮ : then by considering the action of h( s on Ž0, 1, 0. g ⺟ 00 we see that the corresponding action of h( s on J is nontrivial. Therefore all 3 of ²Tt, 0 , S␮ : : Aut ⺟ 00 acts faithfully on J. This establishes an isomor3 phism between the group ²Tt, 0 , S␮ : acting on ⺟ 00 and the group ²Tt, 0 , S␮ : acting on J. We will henceforth assume this identification.

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The following theorem provides the justification for our assertion that the good elements of J are the most important ones and provides the link between ⺟ 03 and the structure of the Jordan algebra. THEOREM 4.3. Suppose X g J y  04 is ␭-potent and has trace ␭. Then Ži. under G, X is equi¨ alent to a real matrix, and Žii. precisely one of X and yX is good. Furthermore, Žiii. e¨ ery nilpotent of J has trace 0. Proof. By the ␲-equivariance of the actions of O Ž ⌿ . on J and ⺟ 03 and 3 those of ² H 15, S␮ : on J and ⺟ 00 , the image under G of a good element is good. Therefore we seek a transform X X of X under G such that just one of "X X is good. Suppose X s Ž a, b, c, u, ¨ , w .. If X Jordan-commutes with both B and C then by Table 4.1 we have X s ␭ A for some ␭ g ⺢, obviously a real matrix. If ␭ ) 0 then X s ␲ Ž'␭ , 0, 0. is good but yX is not. If ␭ - 0 then the reverse applies. Since X / 0 we do not need to consider the case ␭ s 0. If X fails to Jordan-commute with at least one of B and C then we may suppose Žif necessary applying R g G to exchange B with C . that X fails to Jordan-commute with B. If c s 0 then in order for X to be ␭-potent we must have < w < 2 s ␭ c s 0, so w s 0. Even if c / 0, we may suppose without loss of generality that w s 0, as follows. After applying suitable S␮’s to X we may take w g ⺢. Computation shows that Tt, 0 Žfor t g ⺢. acts on J by Tt , 0 : Ž a, b, c, u, ¨ , w . ¬ Ž ?, ?, ?, ?, ?, w q ct . , where the question marks indicate irrelevant coordinates. Applying Tyw r c, 0 we may take w s 0. Since the S␮ and Tt, 0 fix B, we know that X still fails to Jordan-commute with B. Squaring the matrix X we find that when w s 0 the conditions for X to be ␭-potent are aŽ a y ␭. s 0

Ž 4.11.

Ž a q u y ␭. ¨ s 0

Ž 4.12.

¨c s 0

Ž 4.13.

u Ž u y ␭ . q bc s 0

Ž 4.14.

c Ž 2 Re u y ␭ . s 0

Ž 4.15.

< ¨ < 2 q b Ž 2 Re u y ␭ . s 0.

Ž 4.16.

If c s 0 then by Ž4.14., u s 0 or u s ␭, the latter condition being impossible by the trace condition on X. Therefore u s 0, which together with w s c s 0 shows that X Jordan-commutes with B. This contradicts our hypothesis on X, so c s 0 is impossible.

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Since c / 0, Ž4.13. and Ž4.15. show ¨ s 0 and Re u s ␭r2. The trace condition implies that a s 0, and we solve for b by finding b s uŽ ␭ y u.rc s uurc. We conclude 0 Xs 0 0



0 u c

0 uurc . u

0

If c ) 0 then X s ␲ Ž0, 'c , ur'c . and yX is not good. If c - 0 then yX s ␲ Ž0, 'y c , ur'y c . and X is not good. This proves Žii.. To prove Ži., we suppose without loss of generality that c ) 0 and simply apply T0, yIm u r c , which carries X to ␲ Ž0, 'c , Re ur 'c ., a real matrix. To prove Žiii. we suppose that X is nilpotent but make no assumption about its trace. As above, we may suppose w s 0, so that Ž4.11. ᎐ Ž4.16. hold with ␭ s 0. Then Ž4.11. implies a s 0. If bc s 0 then by Ž4.14. we have u s 0, so Tr X s 0. If bc / 0 then by Ž4.13. we find ¨ s 0 and then by Ž4.16. we find Re u s 0, which again shows Tr X s 0. One consequence of Theorem 4.3 is that if X g J is nilpotent then we 3 know that X s "␲ Ž ¨ . for some ¨ g ⺟ 00 with < ¨ < 2⌿ s 0. It is easy to enumerate all possibilities for ¨ , and we find that either X is a multiple of B s ␲ Ž0, 0, 1. or else X s h ⭈ ␲ Ž x, 1, y< x < 2r2 q z . for some h g ⺢, x g ⺟, and z g Im ⺟. THEOREM 4.4. Ži. For each ␭ g ⺢, G acts transiti¨ ely on the good ␭-potents of J. Žii. Two orthogonal nilpotents are proportional. Žiii. The subgroup H 15 of G stabilizes B and acts simply transiti¨ ely on the nilpotents of any gi¨ en nonzero height. Živ. The groups G and Aut J coincide. Žv. The subgroup fixing B and also C is isomorphic to Spin 7 ⺢, is generated by the S␮ , has center  1, RX 4 , and acts on ² B, C, U : as SO Ž7.. Žvi. The stabilizer of B is the semidirect product H 15 i Spin 7 ⺢. Recall that the transformation RX is defined in Ž4.6., as RX : Ž a, b, c, u, ¨ , w . ¬ Ž a, b, c, u, y¨ , yw . . Proof. Ži. Suppose that X and Y are good ␭-potents and thus nonzero. By Theorem 4.3, after applying elements of G, we may suppose that X s ␲ Ž ¨ x . and Y s ␲ Ž ¨ y . with ¨ x , ¨ y g ⺢ 3 y  04 : ⺟ 03 . The norm map 2 ¨ ¬ < ¨ < ⌽ is the usual norm on ⺢ 3 associated to the quadratic form ⌿. We

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know that < ¨ x < 2⌿ s < ¨ y < 2⌿ s ␭ and that O Ž ⌿ . s OqŽ ⌿ . =  "I 4 acts transitively on the nonzero vectors in ⺢ 3 with any given norm. Applying an element of OqŽ ⌿ . : G we may take ¨ x s "¨ y and so X s ␲ Ž ¨ x . s ␲ Ž ¨ y . s Y. Žii., Žiii. For any nilpotent X, either X or yX is good Žthis follows from Theorem 4.3.. In order to have inner product h / 0 with B, we must have X s h ⭈ ␲ Ž x, 1, y< x < 2r2 q z . with x g ⺟, z g Im ⺟. For h / 0 these are obviously permuted simply transitively by H 15, proving Žiii.. If h s 0 then X must have the form "␲ Ž x, 0, z ., but then since 0 s <Ž x, 0, z .< 2⌿ s < x < 2 , we must have x s 0 and so X is a multiple of B. This proves Žii.. Živ. Suppose ␾ g Aut J. By Ži. and Theorem 4.3Žii., Žiii., after multiplying ␾ by an element of G we may suppose ␾ Ž B . s ␧ B, with ␧ s "1. By Žii. and Žiii. we may then take ␾ Ž C . s ␧ X C for some ␧ X g ⺢. Since 2 ␾ Ž B .) ␾ Ž C . s ␾ ŽU1 . must be idempotent we see ␧ X s 1r␧ s ␧ . We will eventually learn that ␧ s q1. The rest of the proof is mostly a chase through the multiplication table of J ŽTable 4.1.. By considering B)C we find that ␾ ŽU1 . s U1. We know that ␾ fixes the identity matrix I Žsince I is the identity element of J ., so it also fixes I y U1 s A. We know that ␾ fixes each of the spaces ² B, C, U :, ² A, B, Im U, V :, and ² A, C, Im U, W :, as these are the subspaces of J which Jordan-commute with A, B, and C, respectively. Taking their intersection we see that ␾ stabilizes Im U. If x g Im ⺟ then x 2 g ⺢ and Ux )Ux s x 2 ⭈ U1 , which shows that ␾ preserves the norm Ux ¬ < x < 2 on Im U. Next, for any x g ⺟, Vx Jordan-commutes with B, so we see that ␾ Ž Vx . s ␣ A q ␤ B q VxX q UyX for some ␣ , ␤ g ⺢, xX g ⺟, and yX g Im ⺟. Consideration of U1 )Vx shows that ␣ s ␤ s yX s 0 and therefore ␾ preserves V : J. Considering Vx )Vx shows that ␾ preserves the norm Vx ¬ < x < 2 on V. Similar reasoning proves that ␾ preserves W : J and the norm Wx ¬ < x < 2 thereon. We conclude that

␾ Ž a, b, c, u, ¨ , w . s Ž a, ␧ b, ␧ c, ␾ 1 Ž u . , ␾ 2 Ž ¨ . , ␾ 3 Ž w . . , with ␧ s "1, each ␾m an orthogonal transformation, and ␾ 1Ž1. s 1. The transformation S␮ acts on Im U by the product of central inversion and reflection in ␮. Therefore ² S␮ : Žthe group generated by all the S␮ . acts on Im U as SO Ž7. and we may suppose that ␾ 1 is either the identity or the conjugation map. ŽWe will see soon that the latter is impossible.. Considering Vx )Wy and using the fact that ␾ 1Ž x . s ␾ 1 Ž x . , we find

␾ 1 Ž xy . s ␾ 2 Ž x . ␾ 3 Ž y .

᭙ x, y g ⺟.

Ž 4.17.

Taking x s y s 1 shows that ␾ 2 Ž1. s ␾ 3 Ž 1 . and we write ␰ for this common value. Taking x s 1 in Ž4.17. yields

␾ 3 Ž y . s ␰␾ 1 Ž y .

Ž 4.18.

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483

and taking y s 1 in Ž4.17. yields

␾ 2 Ž x . s ␾ 1Ž x . ␰ .

Ž 4.19.

Plugging these into Ž4.17. we obtain

␾ 1 Ž xy . s ␾ 1 Ž x . ␰ ⭈ ␰␾ 1 Ž y .

᭙ x, y g ⺟.

Ž 4.20.

If ␾ 1 is the conjugation map then we derive yx s x ␰ ⭈ ␰ y

᭙ x, y g ⺟,

which is impossible since there are noncommuting x and y in some associative algebra containing ␰ . Therefore ␾ 1 s I and so by Ž4.20. we find xy s x ␰ ⭈ ␰ y

᭙ x, y g ⺟.

Taking z g ⺟ and x s ␰ z we may apply the identity Ž2.3. to deduce Ž ␰ z . y s ␰ Ž zy . for all y, z g ⺟. Therefore ␰ g ⺢, so ␰ s "1 and thus by Ž4.18. and Ž4.19. we have ␾ 2 s ␾ 3 s "I. After applying the square RX of any S␮ we may suppose that ␾ 2 s ␾ 3 s I. Finally, consideration of B)Wx shows that ␾ 3 s ␧␾ 2 , so ␧ s q1. Therefore ␾ is the identity, establishing Živ.. We have just seen that the group stabilizing each of B and C is generated by the S␮ , and that this group maps to its action SO Ž7. on ² B, C, U : with kernel equal to  1, RX 4 and central in ² S␮ :. Therefore ² S␮ : is isomorphic to either Spin 7 ⺢ or SO Ž7. = Ž⺪r2.. The latter is impossible because the nontrivial central element RX is a square. This establishes Žv., and Žvi. follows immediately from Žiii. and Žv.. We now define the octave hyperbolic plane ⺟H 2 . It is the image in ⺢ P 26 s PJ of the good negpotents of J. The ‘‘boundary’’ of ⺟H 2 is denoted ⭸ ⺟H 2 and is defined as the image of the good nilpotents. A line in ⺟H 2 is the set of good negpotents that Jordan-commute with some fixed good idempotent. The line associated to a given idempotent X is said to be polar to X. One may define the octave projective plane ⺟P 2 as the image in PJ of all the good elements of J. Then ⭸ ⺟H 2 turns out to be the topological boundary of ⺟H 2 in ⺟P 2 , and each line of ⺟H 2 Žequivalently, each good idempotent of J . is associated with a point of ⺟P 2 y Ž⺟H 2 j ⭸ ⺟H 2 .. We will not need this description of ⺟P 2 . Theorem 4.4 shows that G acts transitively on the points and lines of ⺟H 2 and 2-transitively on the points of ⭸ ⺟H 2 . The remark following Theorem 4.3 shows that ⭸ ⺟H 2 consists of the images of the elements 3 Ž0, 0, 1. and Ž x, 1, z y < x < 2r2. of ⺟ 00 , where x g ⺟ and z g Im ⺟. This 2 realizes ⭸ ⺟H topologically as the one-point compactification S 15 of ⺢ 15 .

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Similarly, ⺟H 2 may be identified with the image of the set Ž x, 1, z y 3 < x < 2r2. : ⺟ 00 N Re z ) 04 . This realizes ⺟H 2 as an open 16-ball bounded 15 by S s ⭸ ⺟H 2 . THEOREM 4.5. G is isomorphic to the group F4Žy 20. , a connected simply connected simple Lie group of 52 dimensions. Proof. The orbit of good nilpotents in J is connected because it fibers over its image S 15 in ⺟H 2 with the fibers being half-lines. The stabilizer of a nilpotent is H 15 i Spin 7 ⺢. This realizes Aut ⺟H 2 as an iterated fibration of connected simply connected spaces and therefore it has these properties itself. We have dim G s 15 q 1 q 15 q dim Spin 7 ⺢ s 52. We may also compute the homological dimension Žh.d.. of G, as h.d. Ž G . s h.d. Ž S 15 . q h.d. Ž ⺢q . q h.d. Ž H

15

. q h.d. Ž Spin 7 ⺢ . s 36.

To show that G is semisimple and centerless it suffices to show that it has no nontrivial normal solvable subgroups. Suppose N were such a subgroup. It is easy to verify that the nilpotents of J span J and this easily implies that N acts faithfully on ⭸ ⺟H 2 . As a normal subgroup of the 2-transitive group G, it acts transitively on ⭸ ⺟H 2 . ŽThis holds because G permutes the orbits of N; if there were more than one then this would contradict the 2-transitivity of G.. This realizes S 15 as the coset space NrM for some subgroup M of N which of course is also solvable. Since solvable groups are aspherical, the long exact homotopy sequence shows that S 15 is aspherical, which is absurd. As a connected centerless semisimple Lie group, G is a direct product of simple Lie groups. Each factor, being normal, must act transitively on S 15 . If M and N are distinct Žthus commuting. factors then since N is transitive, the action of g g M on S 15 is completely determined by its action on any point thereof. This implies that M acts simply transitively on S 15 and hence is compact. If there were more than one factor than this argument would apply to each, exhibiting G as a product of compact groups. Since G is noncompact, this is absurd. Thus there is only one factor, and G is simple. The dimension of G and its simplicity show that it has type F4 . The dimension of any maximal compact subgroup equals h.d.Ž G ., so we see that G is isomorphic to F4Ž52y2 ⭈36. s F4Žy20. . One can see that the stabilizer G 0 of a point of ⺟H 2 is compact, as follows. We know dim G 0 q dim ⺟H 2 s dim G, so dim G 0 s 36. We also know that h.d.Ž G 0 . q h.d.Ž⺟H 2 . s h.d.Ž G ., so h.d.Ž G 0 . s 36. The compactness of G 0 follows from the equality dim G 0 s h.d.Ž G 0 .. One can also show Žsee w16x. that G 0 ( Spin 9 ⺢. We will not need either of these facts.

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5. THE REFLECTION GROUPS A reflection is a nontrivial transformation that fixes a line Žits mirror. of ⺟H 2 pointwise. For each line L there is a unique reflection across L. One can see this by considering the line L polar to A s ␲ Ž1, 0, 0., which contains B s ␲ Ž0, 0, 1. and C s ␲ Ž0, 1, 0.. By part Žv. of Theorem 4.4, the stabilizer of B and C is Spin 7 ⺢, which acts on ² B, C, U : and hence on L as SO Ž7.. Therefore the central element of Spin 7 ⺢ is the only reflection across L. Explicitly, this reflection RX acts on J by RX : Ž a, b, c, u, ¨ , w . ¬ Ž a, b, c, u, y¨ , yw . , which may be written more simply on ⺟ 03 as RX : Ž x, y, z . ¬ Ž yx, y, z . . Since G acts transitively on lines of ⺟H 2 , the reflections form a conjugacy class in G. Coxeter w9x discovered a natural discrete subring K of ⺟. One description w7, p. 14x of K is as the set of vectors Ý x a e a with all x a g 12 ⺪ such that the set of a for which x a g ⺪ q 21 coincides with one of the sets

 0 1 2 44 ,  0 2 3 54 ,  0 1 5 64 ,  0 3 4 64 ,  ⬁ 0 1 34 ,  ⬁ 0 26 4 ,  ⬁ 0 4 5 4 ,  ⬁ 0 1 2 3 4 5 6 4 , or one of their complements. We summarize here the properties of K that we will need. LEMMA 5.1. Ži. As a lattice, K is isometric to a scaled copy of the E8 root lattice, with minimal norm 1, co¨ ering radius 1r '2 and 240 units. Žii. The elements of K of e¨ en norm span K. Žiii. The deep holes of K nearest 0 are the hal¨ es of the elements of K of norm 2. Živ. As a lattice, Im K is a scaled copy of the E7 root lattice, with minimal norm 1, co¨ ering radius 3r4 , and 126 units. Žv. All deep holes of the E7 lattice are equi¨ alent under translations by elements of E7 . Žvi. The group of transformations of Im K generated by the maps x ¬ ␮ x ␮ with ␮ a unit of Im K is the full rotation group of the ⺪-lattice Im K. Žvii. The group of transformation of K generated by the maps x ¬ ␮ x, with ␮ a unit of Im K, is isomorphic to Spin 7 Ž⺖ 2 . and acts transiti¨ ely on the elements of K of each norm 1 and 2.

'

486

DANIEL ALLCOCK

Remarks. Parts Žvi. and Žvii. identify the action of ² S␮ : ␮ is a unit of 3 Im K 4: on ⺟ 00 and hence on J. We will write Spin 7 Ž2. for Spin 7 Ž⺖ 2 . and otherwise use ATLAS notation w7x for finite groups. The deep holes of a lattice L in a Euclidean space are the points of the space furthest from L; the distance from any one of these to L is called the covering radius of L. Proof. Background information on E7 and E8 sufficient to prove Ži., Žiii., Živ., and Žv. is provided in w8, Chap. 4x. It is a simple exercise to prove Žii.; in fact K is the integral span of its elements of norm 2. The map in Žvi. acts on Im K by the product of the real reflection in ␮ and the central involution. Therefore these maps generate the rotation subgroup of the E7 Weyl group, which is the full rotation group of the lattice E7 and hence of Im K. This establishes Žvi.. The central quotient of the group generated by the transformations of Žvii. is identified on p. 85 of w7x as the simple group O 7 Ž2., and since the central involution is a square Žthe square of any S␮ ., the group must be the Žunique. nontrivial central extension Spin 7 Ž2. of O 7 Ž2.. To prove transitivity on the 240 units of K, observe that each orbit must have at least 126 members, so any two orbits meet. Now we show transitivity on norm 2 vectors. Every norm 2 element of K has the form a q b where a and b are orthogonal units of K. By transitivity on units we may take a s 1. Then since Spin 7 Ž2. contains G 2 Ž2. s Aut K Žsee w7, p. 14x., which fixes a s 1 and acts transitively on the units of Im K, we may take b s i Žsay.. This establishes Žvii.. For n a positive integer we define K n as the integral span in J of nA, B,C, and those Ux , 'n Vx , and 'n Wx with x g K. It is easy to see that K is the integral span of the image under ␲ of

 Ž x, y, z . g ⺟03 N x g 'n

⭈ K, y g K, z g K 4

 Ž x, y, z . g ⺟003 N x g 'n

⭈ K , y g ⺪, z g K 4 .

or of

In particular, we can show that an element of G preserves K n if it preserves either of these subsets of ⺟ 03 . We will regard K n as a sort of integral form of J, even though K n is not closed under ). By Aut K n we indicate the subgroup of Aut J preserving the lattice K n and by R n we indicate the subgroup generated by reflections. See Section 7 for an interpretation of the groups Aut K n that make this construction seem natural. We will see that R n s Aut K n if n s 1 or n s 2. Recall that RX acts on ⺟ 03 by RX : Ž x, y, z . ¬ Žyx, y, z ., so we see that X R g R n for all n. The transformation R of Ž4.3. acts on ⺟ 03 by R: Ž x, y, z . ¬ Ž x, yz, yy ., fixing Ž x, y, yy .4 pointwise. Since this subset of ⺟ 03 corresponds to the line polar to ␲ Ž0, 1, 1., we see that R is a reflection. It is also clear that R g R n for all n. The reflection groups we construct

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will be generated by conjugates by certain translations of R and RX . We first show that R n contains a generous supply of translations. THEOREM 5.2. For e¨ ery n and e¨ ery ␰ g K, there exists ␩ g Im ⺟ such that R n contains the transformation T␰ 'n , ␩ . Proof. By Ž4.8. and Lemma 5.1Žii. it suffices to prove the theorem when 3 < ␰ < 2 is even. Computations in ⺟ 00 show that Ty ␰ 'n r2, 0 ( RX (T␰ 'n r2, 0 preserves K n . ŽThis holds even though T␰ 'n r2, 0 might not preserve K n .. Since it is a reflection, it lies in R n . Therefore T␰'n , 0 s RX (Ty ␰'n r2, 0 ( RX (T␰'n r2, 0 lies in R n . Remark. The geometric picture behind this proof is that RX and its conjugate by T␰ 'n r2, 0 are reflections whose mirrors are parallel at infinity Žthat is, their mirrors do not meet in ⺟H 2 , but both contain the point ␲ Ž0, 0, 1. of ⭸ ⺟H 2 .. Naturally, the product of reflections in parallel mirrors is a translation. We now study the stabilizer in R n of nA. Let Jy denote the subspace of J whose elements Jordan-commute with nA. We have Jys ² B, C, U :, a Jordan subalgebra of J. Since R n contains 1 and RX , which are the only elements of G that fix Jy pointwise, the stabilizer in R n of nA is completely determined by its action on Jy. In particular, it must stabilize Jyl K n , which is independent of n and which we denote by Ky. We write Ry for the subgroup of R n generated by all the reflections T0, ␩ ( R(T0, y ␩ with ␩ g Im K. ŽFor each n, we have ² R, T0, ␩ : : Aut K n , so Ry: R n .. Since R and each T0, ␩ fixes A, we see that Ry acts on Jy and Ky. We write ⺟H 1 Žresp. ⭸ ⺟H 1 . for the intersection of ⺟H 2 Žresp. ⭸ ⺟H 2 . with the image of Jy in PJ s ⺢ P 26 . One can identify ⺟H 1 with the real hyperbolic space H 8 and show that its stabilizer in G is SpinŽ8, 1., acting on each of Jy and H 8 as SO Ž8, 1.. ŽIn fact this falls out of the proof of Theorem 5.3 below.. Therefore Ry will act as a group of isometries of H 8. We can describe this group explicitly: THEOREM 5.3. Ži. The action of Ry on ⺟H 1 ( H 8 is that of the rotation subgroup of the real hyperbolic reflection group with Coxeter diagram 䢇

⌬ s

䢇

䢇

䢇

䢇

䢇

䢇

␣

䢇

␤ 䢇

Žii. For each ␩ g Im K and each unit ␮ of Im K, the transformations T0, ␩ and S␮ lie in Ry Ž and hence in R n , for all n.. Žiii. For each n, the stabilizers of B in R n and in Aut K n coincide.

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Remarks. The labels ␣ and ␤ on the diagram are for reference by the proof. This proof is lengthy and the reader may be satisfied with a weaker version of Theorem 5.4, for which this theorem is not needed. Namely, knowing only that the stabilizer of B in R n has finite index in its stabilizer in Aut K n , one can slightly modify the proof of Theorem 5.4 to prove that R n has finite index in Aut K n . ŽWe indicate there how to make these modifications.. To prove this property of the stabilizers of B one need only take the translations of Theorem 5.2 together with those central translations which are their commutators. Proof. The elements of ⭸ ⺟H 1 are the images in projective space PJ of Ž ␲ 0, 0, 1. and those ␲ Ž0, 1, z . with z g Im ⺟. We denote these points by ⬁ and z, respectively, and we will describe symmetries of ⺟H 1 by their actions on Im ⺟ j  ⬁4 . We know that Aut Jy contains the transformations R and T0, ␩ Ž␩ g Im ⺟.. These act as R: z ¬ 1rz T0, ␩ : z ¬ z q ␩ . These formulas also describe the action of R and T0, ␩ on ⺟H 1 s  z g ⺟ N Re z ) 04 . Note that the octave reflection R acts as the central inversion about 1 g ⺟H 1. These transformations generate the conformal group of ⭸ ⺟H 1 s S 7, so this identifies ⺟H 1 with H 8 , as claimed above. We henceforth use the symbol ␩ Žresp. ␮ . exclusively to denote elements Žresp. units. of Im K. By a real reflection we will mean a nontrivial transformation of H 8 that fixes a real hyperplane pointwise. This is the usual definition of reflections, and opposed to octave reflections, which act on H 8 by central inversion in points of H 8. By definition, Rys ² P␩ : where P␩ s T0, ␩ ( R(T0, y ␩ and ␩ varies over Im K. This group is normalized by the involution q: z ¬ z, since q commutes with R and conjugates T0, ␩ to T0, y ␩ . The reader is cautioned that q is not the restriction to Jy of any element of Aut J. Since q reverses orientation on ⭸ ⺟H 1, ² P␩ : is the rotation subgroup of ² P␩ , q :. We write Q␩ for T0, ␩ ( R( q(T0, y ␩ . One can check that Q0 is a real reflection which acts on ⭸ ⺟H 1 by inversion in the unit sphere centered at 0, so Q␩ acts by inversion in the unit sphere centered at ␩. The next few constructions are much more easily carried out by geometric rather than symbolic computation. Figure 5.1 should assist the reader. We know that Q0 g ² P␩ , q :. If <␩ < 2 s 1 then P␩ ( Q0 ( P␩ is the Žreal. reflection across the P␩-translate of the mirror of Q0 ᎏnamely, the perpendicular bisector of the segment joining ␩ and 2␩. If ␩1 and ␩ 2 are two units of Im K with mutual distance 1, then

Ž P␩ ( Q0 ( P␩ .Ž P␩ 1

1

2

( Q0 ( P␩ 2 .Ž P␩ 1 ( Q0 ( P␩ 1 .

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FIG. 5.1. A 2-dimensional section of Im ⺟ Žsee the Proof of Theorem 5.3.. Dots indicate elements of Im K. The real reflection Q0 acts by inversion in the solid circle. The lines are the mirrors of certain other real reflections, whose names are next to them. Each dashed circle is carried to itself by the octave reflection whose name is next to it; this reflection carries each point on the dashed circle to its antipode on the same dashed circle. The octave reflection also exchanges the center of the dashed circle with ⬁.

is reflection across the hyperplane of Im ⺟ that passes through 0 and is orthogonal to "Ž␩1 y ␩ 2 .. These reflections generate the E7 Weyl group, the group of all isometries of Im K fixing 0, and so ² P␩ : contains all of the rotations of Im K fixing 0. Since ² P␩ : is normalized by the T0, ␩ , it contains all the rotations about all of the points of Im K. These rotations generate an Žinfinite . group of transformations of Im ⺟ containing all rotations and translations preserving Im K. We are now in a position to prove Žii.. We know that Ry contains transformations acting on Jy in the same manner as do the T0, ␩ and S␮ . Fixing ␮ for the moment, we see that Ry contains either S␮ or RX ( S␮ . The square of either of these is RX , so RX g Ry. Now, for any ␮ Žresp. ␩ ., Ry contains either S␮ or RX ( S␮ Žresp. T0, ␩ or RX (T0, ␩ . and hence contains both. This proves Žii.. We now finish the proof of Ži.. We know that ² P␩ , q : contains Q0 and all the translations z ¬ z q ␩ , and so it contains all the Q␩ . Since

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q s Q0 ( P0 , we see that ² P␩ , q : s ² P␩ , Q␩ : and so ² P␩ : is the rotation subgroup of ² P␩ , Q␩ :. We now show ² P␩ : : ² Q␩ :. If ␩1 and ␩ 2 are neighbors in Im K then Q␩1 ( Q␩ 2 ( Q␩ 1 is the real reflection in the perpendicular bisector of the segment joining ␩1 and ␩ 2 . These real reflections generate the group of all isometries of Im ⺟ that preserve Im K. ŽThis is just the affine E7 Weyl group; note that the diagram automorphism of the Coxeter group does not preserve the latticeᎏit exchanges it with the set of its deep holes.. In particular, it contains the map q: z ¬ yz. Since P0 s q( Q0 , we see that P0 g ² Q␩ :. Since ² Q␩ : is normalized by all T0, ␩ , we see that ² P␩ : : ² Q␩ :. This identifies ² P␩ : as the rotation subgroup of ² Q␩ :. It remains to identify ² Q␩ :. We take the nodes of ⌬ other than ␣ and ␤ to represent standard generators of the E7 Weyl group, acting on Im K by isometries that preserve 0. We take the node ␣ to represent the reflection in the perpendicular bisector of the segment joining 0 and ␩ 0 , for some unit ␩ 0 of K. The group generated by these 8 reflections is the affine E7 Weyl group and contains all translations of Im K. Taking the node ␤ to represent the reflection Q0 , we see that the group generated by these 9 reflections contains all the Q␩ and thus equals ² Q␩ :. It is easy to see that the angles between pairs of these 9 mirrors are ␲r3 Žresp. ␲r2. when the corresponding nodes of ⌬ are joined Žresp. unjoined.. Since these are integral submultiples of ␲ , the region in H 8 bounded by the mirrors is a fundamental domain for the group generated by the 9 reflections. This identifies ² Q␩ : as the Coxeter group with diagram ⌬, proving Ži.. Now we prove Žiii.. Suppose ␾ g Aut K n with ␾ Ž B . s B. Then ␾ Ž C . is a good nilpotent of height 1, so ␾ Ž C . s ␲ Ž x, 1, z . for some x, z g ⺟. Since ␾ Ž C . g K n we must have x g 'n ⭈ K. After applying a translation of R n , courtesy of Theorem 5.2, we may take x s 0. Then since ␾ Ž C . is nilpotent and in K n we must have z g Im K. After applying T0, yz , which lies in R n by Žii., we may take z s 0. That is, we may suppose ␾ Ž C . s C. By mimicking the proof of Živ. of Theorem 4.4 and using Žvi. of Lemma 5.1 we see that the simultaneous stabilizer of B and C is generated by the S␮ , which by Žii. also lie in R n . This establishes Žiii.. Remarks. The proof of Žii. shows that Ry is a nontrivial central extension by ⺪r2 of its image in Aut Jy; the nontrivial central element is RX . One can show that ⌬ is the Coxeter diagram of the reflection group of the real Lorentzian lattice E7 [ 01 10 . This is not surprising since this form is the negative of the restriction to the traceless elements of Ky of the norm form on J.

ž /

Here is the main result of the paper: the existence of finite covolume reflection groups acting on ⺟H 2 .

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THEOREM 5.4. Ži. Under the action of R1 , there is a single orbit of primiti¨ e good nilpotents of K 1. Žii. Under the action of R 2 , there are precisely two orbits of primiti¨ e good nilpotents of K 2 . Elements of one orbit are characterized as such by being orthogonal to idempotents of K 2 . Žiii. For n s 1 or 2, R n coincides with Aut K n and has finite co¨ olume in G ( F4Žy 20. . Proof. Let n s 1 or 2. Suppose X g K n is a primitive good nilpotent that has minimal height h among its images under R n . Then either X s B or X s h ⭈ ␲ Ž x, 1, y< x < 2r2 q z . with h ) 0, x g ⺟, and z g Im ⺟. After applying a translation in R n Žof which there are plenty by Theorem 5.2., we may suppose that x is at least as close to 0 as it is to any other element of 'n ⭈ K. By Theorem 5.3, R n also contains all central translations, so we may also suppose that z is at least as close to 0 as it is to any other point of Im K. By Lemma 5.1Ži., Žii., these conditions imply < x < 2 F nr2 and < z < 2 F 3r4. Observe that R Ž X . s h ⭈ ␲ Ž x, < x < 2r2 y z, y1 . has height hŽ< z < 2 q < x < 4r4.. If n s 1 then this is smaller than the height h of X, contrary to our hypothesis on X. Thus we have proven Ži.. If n s 2 then either htŽ RŽ X .. - htŽ X ., contrary to our hypothesis on X, or we have < x < 2 s 1 and < z < 2 s 3r4. Since x is at least as close to 0 as to any other element of '2 ⭈ K, we see that x is a deep hole of '2 ⭈ K. Similarly, since z is at least as close to 0 as to any other element of Im K, we see that z is a deep hole of Im K. By Lemma 5.1Žiii., Žvii. and Theorem 5.3Žii., after applying an element of Spin 7 Ž2. : R 2 we may take x to be any particular deep hole of '2 ⭈ K nearest 0, say x s Ž1 q i .r '2 . By Lemma 5.1Žv. and Theorem 5.3Žii. we may apply some T0, ␩ g R 2 and take z to be any particular deep hole of Im K, say z s Ž i q j q k .r2. This determines X up to the scale factor h, which is determined by the requirement that X be a primitive good element of K 2 . Therefore X s X 0 s ␲ Ž 1 q i , '2 , ␻'2 . , where ␻ s Žy1 q i q j q k .r2. This proves that K 2 has at most two orbits of primitive good nilpotents under R 2 . It is obvious that X 0 is orthogonal to the idempotent E s ␲ Ž0, 1, y␻ . g K 2 and one can show B is not orthogonal to any idempotent of K 2 . ŽThe key is that A f K 2 .. This proves that there are exactly two orbits under R 2 and under Aut K 2 ; in particular, Žii. holds. Now suppose ␾ g Aut K n . By using Ži. or Žii., after multiplying by an element of R n we may suppose that ␾ Ž B . s B. But then by Theorem

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5.3Žiii. we see that ␾ g R n . This establishes the equality R n s Aut K n for n s 1, 2. Because Aut K n is an arithmetically defined subgroup of a real semisimple Lie group, a theorem of Borel and Harish-Chandra w5x shows that it has finite covolume in G, which establishes Žiii.. Remarks. If we do not assume the full results of Theorem 5.3, as described in the remark there, the proof above can be modified to obtain the weaker result that R1 and R 2 have finite covolume in G. The proof above shows that if X is a primitive good nilpotent of K 1 of positive height then there is a reflection in R1 reducing the height of X Žnamely, the conjugate of R by the translations used in the first paragraph of the proof.. Therefore we can conclude that X is equivalent to B under R1. Similarly, if X is a primitive good nilpotent in K 2 then after a sequence of reflections in R 2 , X may be taken to have height F 2. Since the stabilizer of B in Aut K 2 acts with one orbit on the primitive good nilpotents of height 1, and by hypothesis the stabilizer of B in Ry: R 2 has finite index of that in Aut K 2 , we see that there are only finitely many R 2-orbits of primitive good nilpotents in K 2 . The finiteness of the index of R i in Aut K i for each i s 1, 2 follows from this and from the assumption that B’s stabilizer in R i has finite index in its stabilizer in Aut K i . 6. THE DETERMINANT In Section 5 we constructed the promised octave hyperbolic reflection groups and proved that they have finite covolume. In the following section we will interpret our groups as the stabilizers of certain Hermitian forms over K. To do this we must work in a context wider than that of our treatment so far. Namely, we must introduce the determinant form on J and its automorphism group H. By explicit construction we will show that the determinant form on J is equivalent to the one on JI studied in w11x, which will prove that H ( E6Žy 26. . The action of E6Žy26. on JI is similar to the action of PGL3 ⺓ on the space of Hermitian forms on ⺓ 3, so this is the natural setting for discussions of the stabilizers of Hermitian forms. We will pursue this further in the next section. Recall that we defined ␹ Ž X . s Re TrŽ X . for any X g M3 ⺟. ŽOf course, we are only interested in elements of Jordan algebras J⌽ , which have real traces.. Pages 30᎐31 of w12x show that if X, Y, Z g M3 ⺟ then ² X N YZ : s ² XY N Z : s ² ZX N Y :, and it follows that ␹ Ž X 2 X . s ␹ Ž XX 2 . and that ² X < Y < Z : s ² X )Y N Z : is a symmetric trilinear form. Since it is symmetric, it is obtained by polarization from the cubic form C given by C Ž X . s ² X < X < X : s ␹ Ž X 3. .

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Following w11x we define the cubic ‘‘determinant’’ form by det Ž X . s

1 6

3

2 ␹ Ž X 3 . y 3␹ Ž X 2 . ␹ Ž X . q ␹ Ž X . .

The name of this cubic form derives from analogies between it and the usual determinant form on Žsay. M3 ⺢. In particular, in the special cases will consider below, the formulas Ž6.1. and Ž6.2. for the determinant closely resemble the usual expression. For any ⌽ we write H⌽ for the group of all linear transformations preserving the determinant form on the real vector space underlying J⌽ . We write H for H ⌿ . THEOREM 6.1.

For X g J gi¨ en by Ž4.1.,

det Ž X . s a < u < 2 q b < w < 2 q c < ¨ < 2 y abcy 2 Re Ž u¨ w . .

Ž 6.1.

Proof. The computation is long and tedious but mostly straightforward. One should begin by writing 2 TrŽ X 3 . s TrŽ X 2 X . q TrŽ XX 2 . before multiplying together the matrices. This device makes the resulting sum simplify because many of its terms appear in ⺟-conjugate pairs. Manipulations involving no special ⺟ identities except obvious applications of Ž2.2. show det Ž X . s a < u < 2 q b < w < 2 q c < ¨ < 2 y abcq 2 Re Ž u¨ w . y 4 Re Ž ¨ w . Re Ž u . , and so we examine the term in brackets. Further use of Ž2.2. allows the derivation 2 Re Ž u¨ w . y 4 Re Ž ¨ w . Re Ž u . s Re 2 Re Ž u¨ w . y 4 Re Ž ¨ w . Re Ž u . s Re u¨ w q w ¨ u y Ž ¨ w q w ¨ . Ž u q u . s Re u¨ w q w ¨ u y ¨ wu y ¨ wu y w ¨ u y w ¨ u s Re uw ¨ q uw ¨ y uw ¨ y u¨ w y u¨ w y uw ¨ s y2 Re Ž u¨ w . , which completes the proof. Since the action of GL3 ⺢ on M3 ⺟ preserves matrix multiplication and traces, it also preserves the determinant form. In light of this, the next theorem assures us that G : H; it also provides a quick alternate proof of most of Theorem 4.1.

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THEOREM 6.2. The transformations S␮ preser¨ e the restriction to J of the determinant form and act as automorphisms of the Jordan algebra J. Proof. To show that det S␮Ž X . s det X for X g J given by Ž4.1. it suffices to show Re Ž ␮ u ␮ ⭈ ␮¨ ⭈ w␮ . s Re Ž u¨ w . . This follows from the derivation Re Ž ␮ u ␮ ⭈ ␮¨ ⭈ w␮ . s yRe Ž ␮ u ␮ ⭈ ␮¨ ⭈ w␮ . s yRe Ž ␮ u ␮ ⭈ ␮ Ž ¨ w . ␮ . s yRe Ž ␮ u ␮ ⭈ ␮ ⭈ Ž ¨ w . ␮ . s yRe Ž ␮ u ␮␮ ⭈ Ž ¨ w . ␮ . s yRe Ž ␮ u ⭈ Ž ¨ w . ⭈ ␮ . s yRe Ž ␮ ⭈ ␮ u ⭈ ¨ w . s Re Ž u¨ w . , where we have used Ž2.1., Ž2.2., and the fact that ␮ s y␮. ŽThis is in spirit the same argument as that applied to JI in w11, Sect. 6x.. Combined with the fact that the S␮ preserve traces and norms, this proves that S␮ g Aut J. This follows because the Jordan multiplication can be defined in terms of the trace, norm, and determinant forms: Suppose X, Y g J. Supposing that we know the trace, norm, and determinant forms, we know the value of C for every element of J. Therefore we know the value of the symmetric trilinear form ² X < Y < Z : obtained from C by polarization, for every Z g J. By the definition of ² X < Y < Z :, we know the value of ² X )Y < Z : for every Z, and since the inner product is nondegenerate this uniquely determines X )Y. There are elements of H that do not lie in G, for example, if r g ⺢ y  04 then H contains the map Fr : Ž a, b, c, u, ¨ , w . ¬ Ž ar 4 , bry2 , cry2 , ury2 , ¨ r , wr . . We may identify H with the real Lie group E6Žy 26. as follows. In w11x Freudenthal calculated the determinant form on JI . An element of JI has the form



d z y

z e x

y x , f

0

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for d, e, f g ⺢ and x, y, z g ⺟. We denote this element of JI by w d, e, f, x, y, z x. It is shown in w11x that det w d, e, f , x, y, z x s def q 2 Re Ž xyz . y d < x < 2 y e < y < 2 y f < z < 2 , Ž 6.2. and that the HI is isomorphic to E6Žy26. . To identify H with this Lie group we need only observe that the map g: J ª JI defined by g : Ž a, b, c, u, ¨ , w . ¬ w ya, yc, yb, yu, y¨ , yw x identifies the determinant forms. So far in this paper we have informally regarded elements of a Jordan algebra J⌽ as ‘‘transformations of ⺟ 3 ’’ that are Hermitian with respect to ⌽. When one considers just the determinant form on JI and not the Jordan algebra structure, it is more appropriate to regard elements of JI as ‘‘Hermitian forms on ⺟ 3.’’ One should also regard their determinants as those of Hermitian forms rather than of ‘‘linear transformations.’’

7. STABILIZERS OF INTEGRAL HERMITIAN FORMS We write HIK for the stabilizer in HI of JI l M3 K. Given the material we have developed so far, it is quite easy to realize the groups R n as stabilizers in HIK of appropriate forms over K. One can show that F4Žy 20. is a maximal Žproper. subgroup of E6Žy26. , and since not all of H fixes the identity matrix I, we see that we may define G as the subgroup of H fixing I g J. Recall that for n a positive integer we defined K n s  Ž na, b, c, u, ¨ 'n , w'n . g J a, b, c g ⺪, u, ¨ , w g K 4 . From the above, we conclude that Aut K n is the subgroup of H that preserves K n as a set and fixes nI g K n . We define the map h n : J ª JI by h nŽ X . s yny1 r3 ⭈ Ž g ( Fr .Ž X . where r s ny1r6 and Fr and g are as defined in Section 6. Computation reveals that h n carries K n bijectively to JI l M3 K and multiples determinants by y1rn. This shows that Aut K n is isomorphic to the subgroup of HIK that fixes h nŽ nI .. Computing h nŽ nI . we obtain:

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THEOREM 7.1. For any n, Aut K n is isomorphic to the subgroup of HIK that preser¨ es the matrix Ž or ‘‘Hermitian form’’.

ž

1 0 0

0 0 n

0 n . 0

/

An element g g PGL3 ⺪ acts on 3-dimensional integral quadratic forms ⌽ by ⌽ ¬ g ⌽ g U and obviously acts on JI l M3 K in the same way, realizing PGL3 ⺪ as a subgroup of HIK . The stabilizer in PGL3 ⺪ of a quadratic form ⌽ over ⺪ is Žthe central quotient of. the automorphism group of the lattice with inner product matrix ⌽. This makes it natural and satisfying to regard the groups Aut K n Žand in particular the reflection groups R1 and R 2 . as the symmetry groups of ‘‘Lorentzian lattices’’ over K with the ‘‘inner product matrices’’ given in Theorem 7.1. Gross w13x has investigated integral forms of various semisimple algebraic groups, and observed that the stabilizer in HIK of the matrix ⌿X s

ž

0 0 y1

0 y1 0

y1 0 0

/

is a model over ⺪ for the unique form of F4 over ⺡ which is split at each prime p and has rank 1 over ⺢. Since ⌿X is equivalent to y⌿ under PGL3 ⺪ : HIK we see that this integral form of F4 is the reflection group R1. In particular, one of our groups has been investigated before and found to be a natural integral form of F4 . Finally, one can consider 2-dimensional versions of our constructions. As introduced in Section 5, Jy is the space of 2 = 2 octave matrices which are Hermitian with respect to 01 10 . ŽWe actually introduced Jy in a slightly different but equivalent way.. That is, Jy consists of the matrices ub with b, c g ⺢ and u g ⺟. There are no subtleties involved in the cu definition of the determinant < u < 2 y bc of an element of Jy, and the group preserving just the vector space structure and determinant form on Jy is obviously the real orthogonal group O Ž9, 1.. The group of Jordan algebra automorphisms is the stabilizer O Ž8, 1. therein of the identity matrix. We also defined Ky as M2 K l Jy. The determinant form on the space

ž /

ž /

X Jy s

e x

x f

½ž /

e, f g ⺢, x g ⺟

5

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of matrices in M2 ⺟ that are Hermitian with respect to X ef y < x < 2 . Thus the map Jyª Jy given by

10 01

ž / is given by

ž uc ub / ¬ ž ub uc / negates determinants, carries 10 01 to 01 10 , and identifies Ky with M2 K l X Jy . Therefore, as in the 3-dimensional case, we may interpret the group Aut Ky as being essentially the automorphism group of the ‘‘Lorentzian lattice’’ with inner product matrix 01 10 . In Section 5 we defined a group Ry which acted on Ky; it turned out that Ry was a central extension by ⺪r2 of the rotation subgroup of a certain Coxeter group. The map Ryª Aut Ky is almost an isomorphism. The kernel of the map is the central ⺪r2 of Ry and the image has index two in Aut Ky, which is the entire Coxeter group. ŽThe real reflections of H 8 s ⺟H 1 arise from automorphisms of Jy that do not extend to J.. This description of Aut Ky follows from the fact that the central involution of Ky does not lie in Aut Ky, together with the remark concerning the lattice E7 [ 01 10 following the proof of Theorem 5.3.

ž / ž /

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REFERENCES 1. D. J. Allcock, New complex and quaternion-hyperbolic reflection groups, preprint, 1996. 2. D. J. Allcock, Identifying models of the octave projective plane, Geom. Dedicata 65 Ž1997., 215᎐217. 3. H. Aslaksen, Restricted homogeneous coordinates for the Cayley projective plane, Geom. Dedicata, 40 Ž1991., 245᎐250. 4. R. E. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 Ž1987., 133᎐153. 5. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 Ž1962., 485᎐535. 6. J. H. Conway, The automorphism group of the 26-dimensional even unimodular Lorentzian lattice, J. Algebra 80 Ž1983., 159᎐163. 7. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘ATLAS of Finite Groups,’’ Oxford University Press, Oxford, 1985. 8. J. H. Conway and N. J. A. Sloane, ‘‘Sphere Packings, Lattices, and Groups,’’ SpringerVerlag, New YorkrBerlin, 1988. 9. H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 Ž1946., 561᎐578. 10. N. Elkies and B. H. Gross, The exceptional cone and the Leech lattice, preprint, 1996. 11. H. Freudenthal, Zur ebenen Oktavengeometrie, Neder. Akad. Wetensch. 56 Ž1953., 195᎐200. 12. H. Freudenthal, Oktaven, Ausnahmegruppen, und Oktavengeometrie, Geom. Dedicata 19 Ž1985., 1᎐73. 13. B. H. Gross, Groups over Z, In¨ ent. Math. 124 Ž1996., 263᎐279.

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¨ 14. P. Jordan, Uber eine nicht-Desarguessche ebene projective Geometrie, Abh. Math. Sem. Uni¨ . Hamburg 16 Ž1949., 74᎐76. 15. G. D. Mostow, Strong rigidity of locally symmetric spaces, in ‘‘Annals of Mathematics Studies,’’ Vol. 78, Princeton Univ. Press, Princeton, NJ, 1973. 16. J. Tits, Le plan projectif des octaves et les groups de Lie exceptionnels, Bull. Acad. Roy. Belg. Classe Sci. 39 Ž1953., 300᎐329. 17. J. Tits, Le plan projectif des octaves et les groupes exceptionnels E6 et E7 , Bull. Acad. Roy. Belg. Classe Sci. 40 Ž1954., 29᎐40. 18. E. B. Vinberg, On the groups of units of certain quadratic forms, Math. USSR-Sb. 16, No. 1 Ž1972., 17᎐35. 19. E. B. Vinberg, On unimodular integral quadratic forms, Funct. Anal. Appl. 6 Ž1972., 105᎐111. 20. E. B. Vinberg and I. M. Kaplinskaja, On the groups O18, 1Ž Z . and O19, 1Ž Z ., So¨ iet Math. Dokl. 19, No. 1 Ž1978., 194᎐197.