Jordan Algebras andF4Bundles over the Affine Plane

198, 582]607 Ž1997. JA977161

JOURNAL OF ALGEBRA ARTICLE NO.

Jordan Algebras and F4 Bundles over the Affine Plane R. Parimala,* V. Suresh,† and Maneesh L. Thakur ‡ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400 005, India Communicated by E¨ a Bayer-Fluckiger Received March 10, 1997

INTRODUCTION Let k be a field of characteristic different from 2 and 3. Let J be a finite dimensional exceptional central simple Jordan algebra over k. Let G be the group of automorphisms of J. Then G is an algebraic group of type F4 defined over k. Let X be an integral scheme over k. The set H 1 Ž X, G . classifies the isomorphism classes of principal G-bundles on X. This set also classifies the isomorphism classes of exceptional simple Jordan algebra bundles on X. We assume that G is anisotropic and that the Jordan algebra J is reduced. In this paper, we construct an infinite family J w i x of Jordan algebra bundles over the affine plane A2k which are mutually non-isomorphic and which specialise to J at any rational point. These bundles have the rigidity property that the associated principal G-bundles PJ w i x admit no reduction of the structure group to any proper connected reductive subgroup of G. A theorem of the same kind for principal G-bundles, G connected reductive anisotropic, with the exceptions of G 2 and F4 , was proved by Raghunathan in wRx. The case G s Aut J, with J a division Jordan algebra, will be dealt with elsewhere, thus settling the case of F4 . Let k be as above and C a Cayley algebra over k. Let G s ²g 1 , g 2 , g 3 : be a diagonal matrix with g i g k*. Let s : M3 Ž C . ª M3 Ž C . be the involution given by s Ž X . s Gy1 X t G, bar denoting the entrywise action of the canonical involution on C. Let J s H3 Ž C, G . be the set of symmetric * E-mail address: [email protected]. † E-mail address: [email protected]. ‡ E-mail address: [email protected]. 582 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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elements for s . The multiplication A. B s 12 Ž AB q BA. makes J into a Jordan algebra over k. Every reduced exceptional Jordan algebra is constructed this way. As a sequel to wKPS2x, with the assumption that C is a Cayley division algebra, an infinite family of Cayley algebra bundles Ci on A2k was constructed in wTx with the property that the quadratic bundle CiX of trace zero elements in Ci is indecomposable. Then  Ji s H3 Ž Ci , G .4 is a family of Jordan algebra bundles on A2k which specialize to J at a rational point. Suppose that the group G s Aut J is anisotropic Žcf. Ž3.1... Then C is a division algebra. Using the algebras Ji as prototypes on open subsets of A2k and using patching techniques of wPx, we construct an infinite family of Jordan algebra bundles J w ix over A2k with the following properties Ži. J w ix specialises to J at any point. Žii. The associated principal G-bundle PJ w i x admits no reduction of the structure group to any proper connected reductive subgroup of G. Certain rigidity properties for H-bundles, where H is a connected reductive group, and in particular for the orthogonal bundles, are proved in Section 2, which are used in subsequent sections for proving the rigidity of the G-bundles, with G as above. We remark that, since for k a number field or the field of real numbers, every exceptional simple Jordan algebra over k is reduced, the proof for the case F4 of Raghunathan’s theorem is complete for these fields.

1. RIGID FORMS Let Ž E , q . be a quadratic space over a scheme X. We call q rigid if the orthogonal group of q consists only of " identity. We owe this terminology to W. Scharlau, who called such forms ‘‘starr.’’ In this section we give a characterisation of rigid spaces over the projective plane Pk2 . Let N be an indecomposable vector bundle over Pk2 . Then N s End N is a finite dimensional local k-algebra and D s Nrrad N is a division algebra over k. In fact N is absolutely indecomposable Ži.e., Nk is indecomposable, k denoting the algebraic closure of k . if and only if D s k. Suppose N is self dual and a : N ª N * is an isomorphism. Since N is local, replacing a by a " a t , we may assume that a t s ea with e s "1. The isomorphism a gives rise to an involution t : N ª N given by t Ž f . s a f tay1 which induces an involution t : D ª D. We recall wQSS, Theorem 3.3x the equivalence between the categories of quadratic spaces over Pk2 with underlying vector bundle isotypical of type N and e-hermitian spaces over Ž N, t .. Let E s [l c o p i e s N be a vector bundle on Pk2 with a symmetric isomorphism q : E ª E *. The free N-module E s Hom Ž N , E . sup-

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ports the e-hermitian form q˜ : E ª Hom N Ž E, N . defined by q˜Ž f . Ž g . s ay1 ( g t ( q( f . The assignment Ž E , q . ¬ Ž E, q˜. yields the required equivalence. We also recall wQSS, Theorem 3.3x that every e-hermitian space over Ž N, t . is determined up to isometry by its reduction modulo rad N, which is a space over Ž D, t .. Suppose a : N ª N * is symmetric. Using the above equivalence, combined with reduction modulo rad N, one gets the following LEMMA 1.1. The set of isometry classes of quadratic spaces, with N as the underlying ¨ ector bundle, is in bijection with the set  u g D*
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Proof. Suppose O Ž q . s  "14 . Then clearly q is indecomposable as a quadratic space. By Ž1.2., E is indecomposable as a vector bundle. Let E s End E and D s Errad E and t the involution on E and D, induced by q. If E is not absolutely indecomposable, then D / k. There exists u g D*, u / "1, such that ut Ž u. s 1. Since rad E is nilpotent, u lifts to a unit u ˜ g End E such that u˜t Ž u˜. s 1, i.e., uqu ˜ ˜tqy1 s 1, i.e., u˜ g OŽ q . and u ˜ / "1, contradicting the assumption on q. Thus E is absolutely indecomposable. Suppose conversely that E is absolutely indecomposable. Let u g O Ž q .. Then ut Ž u. s 1. Since Errad E s k and t is identity on k, Ž u. 2 s 1, bar denoting reduction modulo rad E. Replacing u by yu, if necessary, we may assume that u s 1, i.e., u s 1 q h , h g rad E, i.e., Ž u y 1. n s 0 for some n. Thus u, as an element of the orthogonal group of q m k Ž X, Y ., is unipotent. Since q is generally anisotropic, O Ž q m k Ž X, Y .. has no nontrivial unipotent elements wBo, 6.4.2x. Thus u s 1. This completes the proof. COROLLARY 1.4. Let Ž E , q . be an indecomposable anisotropic quadratic space o¨ er Pk2 of rank p, p a prime. Then q is rigid. Proof. By Ž1.2., E is indecomposable as a vector bundle. Suppose that Ek is decomposable. One concludes by looking at the Galois action on the decomposition of Ek into direct sum of isotypical components, that Ek is a direct sum of line bundles. Then clearly E itself is a direct sum of line bundles, contradicting its indecomposability. Thus Ek is indecomposable and by Ž1.3., O Ž q . s  "14 . 2. RIGID FORMS OVER A2k We recall from wKPS, Theorem 2.1x that every anisotropic quadratic space over A2k extends uniquely to a quadratic space on Pk2 . We begin with a corresponding statement on extension of isometries. LEMMA 2.1. Let R be a domain and p a prime in R. Let q and qX be quadratic spaces o¨ er R. Suppose that qX m RrŽ p . is anisotropic. Let a : q m Rw1rp x ª qX m Rw1rp x be an isometry. Then a extends uniquely to an isometry a : q , qX . Proof. Compare wKPS1, Proposition 1.1x. COROLLARY 2.2. Let q˜ be an anisotropic quadratic space o¨ er Pkn and q its restriction to Ank . Then O Ž q . s O Ž q˜.. Proof. Clearly, restriction yields an injection O Ž q˜. ¨ O Ž q .. Let a g O Ž q .. Let X i , 0 F i F n, be the homogeneous coordinates of Pkn such that Ank s DŽ X n .. Let a i denote the restriction of a to DŽ X i . l DŽ X n .. By

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Ž2.1., a i extends uniquely to an isometry a ˜i on DŽ X i .. Let a˜i j denote the restriction of a ˜i to DŽ X i . l DŽ X j .. Then a˜i j s a˜ji on DŽ X i . l DŽ X j . l DŽ X n . by Ž2.1. and hence they coincide. Thus the  a ˜i 4 patch on DŽ X i . l DŽ X j . to yield an isometry a ˜ : q˜ ª q˜ which uniquely extends a . This proves the corollary. PROPOSITION 2.3. Let q be an anisotropic quadratic space o¨ er A2k and Ž E , q˜. its extension to Pk2 . Then q is indecomposable if and only if E is indecomposable as a ¨ ector bundle. Further, q is rigid if and only if E is absolutely indecomposable. Proof. Since the extension Ž E , q˜. of q is unique, if q decomposes then so does q˜ and hence E decomposes as a vector bundle. Conversely, if E decomposes as a vector bundle, by Ž1.2., q˜ being anisotropic, q˜ decomposes and hence q decomposes. Since O Ž q . s O Ž q˜., Ž2.2., the second assertion in the proposition is immediate from Ž1.3.. COROLLARY 2.4. Let q be an indecomposable quadratic space o¨ er A2k of prime rank. Then q is rigid Ž i.e., O Ž q . s  "14.. Proof. This is immediate from Ž1.4. and Ž2.3.. COROLLARY 2.5. Let qi be rigid quadratic spaces o¨ er A2k such that H qi is anisotropic. Suppose qi and q j are not similar for i / j. Then e¨ ery isometry a : H qi ªH qi maps qi to qi . Proof. Since each qi is rigid, if Ž Ei , q˜i . denotes its extension to Pk2 , Ei is absolutely indecomposable. In particular, each Ei supports a unique quadratic space structure up to scalars. Since qi is not similar to q j , q˜i is not similar to q˜j . Hence Ei ` Ej for i / j. By Ž2.2., a extends to an isometry a ˜ : H Ž Ei , q˜i . ªH Ž Ei , q˜i .. By the Krull]Schmidt theorem, a ˜ Ž Ei . s Ei so that a Ž qi . s qi . We conclude this section by giving a construction of rigid spaces over A2k s SpecŽ k w X, Y x.. The idea of this construction goes back to the construction of rigid spaces over A2k given in wPx. Let p1 , p 2 g k w X x be polynomials such that Ž p1 , p 2 . s 1. Let Ž q1 , q2 ., Ž q3 , q4 . be pairs of rigid quadratic spaces over A2k such that q1 , q2 are not similar and q3 , q4 are not similar. Let q0 , qX0 be quadratic forms over k such that q0 H qX0 is anisotropic and q1 s q3 s q0 m k w X x, q2 s q4 s qX0 m k w X x, bar denoting the reduction modulo Y. Suppose that p 1 : Ž q1 . p 1 ª Ž q0 . p 1, p 2 : Ž q2 . p 1 ª Ž qX0 . p are isometries over k w X x p w Y x such that p i s identity, i s 1, 2. 1 1 Suppose p 3 : Ž q2 . p 2 ª Ž q0 . p 2 , p4 : Ž q4 . p 2 ª Ž qX0 . p 2 are isometries over k w X x p 2 w Y x such that p i s identity, i s 3, 4. Let c g O Ž q0 H qX0 . be such that c Ž q0 . and c Ž qX0 . are contained neither in q0 not in qX0 . Let q be the quadratic space over A2k obtained by taking the space Ž q1 H q2 . p 2 over

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K w X x p 2 w Y x and Ž q3 H q4 . p 1 over k w X x p 1w Y x and patching them over the intersection k w X x p 1 p 2 w Y x by the isometry u s Žp 3 H p4 .y1 oc oŽp 1 H p 2 .. PROPOSITION 2.6. The space q, constructed abo¨ e, is rigid. Proof. Any isometry a : q ª q is given by a pair of isometries a 1 : Ž q1 H q2 . p 2 ª Ž q1 H q2 . p 2 over k w X x p 2 w Y x and a 2 : Ž q3 H q4 . p 1 ª Ž q3 H q4 . p 1 over k w X x p 1w Y x such that the diagram u 1 p2

Ž q 3 H q4 . p

6

Ž q1 H q 2 . p a1

6

6

u 1 p2

Ž q 3 H q4 . p

6

Ž q1 H q 2 . p

1 p2

a2 1 p2

commutes. Let bar denote the reduction modulo Y. Then u˜s c is defined over k and a 2 c s ca 1 is defined over both k w X x p 1 and k w X x p 2 . Since Ž p1 , p 2 . s 1, a 1 and a 2 are defined over k w X x. Since q1 H q2 is extended after inverting p1 and a 1 is an isometry of Ž q1 H q2 . p 2 onto itself with a 1 defined over k w X x, by wP, 2.4x, a 1 is defined over k w X, Y x. Since q1 and q2 are rigid and nonsimilar, by Ž2.5., a 1Ž q1 . s q1 and a 1Žq 2 . s q2 . Since the qi are rigid, a 1 s e 1 H e 2 where e i : qi ª qi is given by "1. Similarly a 2 s e 3 H e 4 with e i s "1. Thus a i s a i and we have Ž e 3 H e 4 . c s c Ž e 1 H e 2 .. Writing

cs

ž

c1 c3

c2 c4

/

as a transformation of q0 H qX0 , we have, by the assumptions on c , c i / 0 for 1 F i F 4. Further,

ž

e3 c1 e4 c 3

e3c2 e1c1 s e 4 c4 e1c3

/ ž

e2 c2 e 2 c4

/

from which it is immediate that a s "1. Therefore O Ž q . s  "14 , so that q is rigid.

3. SOME GENERALITIES ON REDUCED EXCEPTIONAL JORDAN ALGEBRAS Let R be a commutative ring in which 2 is invertible. Let Zorn Ž R . denote the R-module Ž RR3 RR3 .. The multiplication

ž

a xX

x aX

/ž

b yX

y ab y x. yX X s X b bx q aX yX q x = y

/ ž

ay q bX x q xX = yX , aX bX y xX . y

/

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where x, y and x = y respectively denote the usual scalar and vector product on R 3, makes Zorn Ž R . into a non-associative R-algebra. This algebra has a canonical involution defined on it, namely, x ¬ A s aX X aX yx

A s aX x

ž

/

yx . a

ž

/

One has ABs BA and nŽ A. [ AA s AA g R, t Ž A. [ A q A g R. The map n is a non-singular quadratic form which is multiplicative, i.e., n Ž AB . s n Ž A . n Ž B . ,

A, B g Zorn Ž R . .

DEFINITION. A Cayley algebra over R is a non-associative R-algebra C such that for some faithfully flat extension SrR, C mR S , Zorn Ž S . as S-algebras. The algebra C acquires a non-degenerative quadratic form n C which we call the norm on C. It also has an involution bar with the property that n C Ž X . s XX s XX g R ,

for all X g C.

If R is a field, n C is a 3-fold Pfister form and the algebra C is division if and only if n C is anisotropic. Let C be a Cayley algebra over R. For g 1 , g 2 , g 3 , units in R, let G denote the diagonal matrix ²g 1 , g 2 , g 3 :. Let H3 Ž C, G . s  A g M3 Ž C . ¬ Gy1At G s A4 , where M3 Ž C . denotes the set of 3 = 3 matrices with entries in C and bar denotes entrywise action of the involution bar on C. Explicitly we have

¡

H3 Ž C, G . s

~



¢

a1

c

gy1 1 g3 b

gy1 2 g1c

a2

a

b

gy1 3 g2 a

a3

0

¦

¥

a, b, c g C, . ai g R

§

We note that H3 Ž C, G . is a projective R-module of rank 27. Let AB denote the usual matrix product in M3 Ž C .. On H3 Ž C, G . we define a multiplication by A. B s 12 Ž AB q BA.. This makes H3 Ž C, G . into a non-associative commutative algebra over R. If R is a field, H3 Ž C, G . is a reduced exceptional simple Jordan algebra and every reduced simple exceptional Jordan algebra is isomorphic to H3 Ž C, G . for suitable C and G. For



a1

A s gy1 2 g1c b

c

gy1 1 g3 b

a2

a

gy1 3 g2 a

a3

0

g H3 Ž C, G . ,

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one defines a trace function by T Ž A. s a 1 q a 2 q a 3 , the sum of the diagonal entries of A. Then QŽ A. s T Ž A2 . is a non-singular quadratic form on H3 Ž C, G ., called the trace form. In fact for A as above, Q Ž A . s T Ž A2 . s a 12 q a 22 q a 32 y1 y1 q 2 Ž gy1 3 g 2 nC Ž a. q g 1 g 3 nC Ž b . q g 2 g 1 nC Ž c . . ,

where n C is the norm form of C. There is a cubic form N defined on H3 Ž C, G . given by y1 N Ž A . s a 1 a 2 a 3 y gy1 3 g 2 a 1 nC Ž a. y g 1 g 3 a 2 nC Ž b .

y gy1 2 g 1 a 3 n C Ž c . q t Ž Ž ca . b . , t denoting the trace on C given by t Ž x . s x q x. Every element A of H3 Ž C, G . satisfies the cubic polynomial equation X 3 y T Ž A. X 2 q

1 2

Ž T Ž A . 2 y T Ž A2 . . X y N Ž A . s 0.

Ž w.

By an exceptional Jordan algebra over R we mean an algebra J such that for a faithfully flat extension SrR, J mR S , H3 Ž C, G . as S-algebras for some C and G defined over S, C, and G as above. Such an algebra carries a norm N and a trace T which are descents of the norm and trace of H3 Ž C, G .. In particular, every element of J satisfies the cubic polynomial equation given by Žw.. In this paper we shall sometimes use the abbreviation Jordan algebras to mean exceptional Jordan algebras. There is an obvious notion of an exceptional Jordan algebra over any algebraic scheme. If k is a field, the group of automorphisms of exceptional simple Jordan algebras are precisely the algebraic groups of type F4 over k. We conclude this section by giving a criterion for the group G s Aut J, J a reduced exceptional simple Jordan algebra over the field k, to be anisotropic. LEMMA 3.1. The following are equi¨ alent Ži. G is isotropic. Žii. J contains non-zero nilpotent elements. y1 y1 : Žiii. J is reduced and the rank 24 subform ²gy1 3 g 2 , g 1 g 3 , g 2 g 1 m nC of Q is isotropic. Živ. J contains a non-zero element a with T Ž a. s QŽ a. s N Ž a. s 0. Proof. Žii. m Živ.. Any element a of J satisfies X 3 y T Ž a. X 2 q

1 2

Ž T Ž a. 2 y T Ž a2 . . X y N Ž a. s 0.

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If a is a non-zero nilpotent then clearly T Ž a. s QŽ a. s N Ž a. s 0. Conversely if Živ. holds then we get a3 s 0 so that a is nilpotent. The equivalence of Žiii. and Žii. follows by wAJ, Theorem 6; J2, Exercise 1, p. 381x. We now show Ži. « Žii.. Since G is isotropic, there exists an injection i : G m ¨ G for a k-split G m . Let J X denote the subspace of J consisting of all trace zero elements of J. Then G m acts faithfully on J X and J X can be written as JX s

26

[ k.¨ i ,

k.¨ i being eigenspaces for G m .

is1

For any i, 1 F i F 26 and t g G m , t Ž ¨ i . s li Ž t . ¨ i for some character l i of G m . Therefore T Ž ¨ i . s li Ž t . T Ž ¨ i . ,

2

Q Ž ¨ i . s li Ž t . Q Ž ¨ i . ,

and

3

N Ž ¨ i . s li Ž t . N Ž ¨ i . . So, if for some i and some t, none of l i Ž t ., l i Ž t . 2 , l i Ž t . 3 is 1, then T Ž ¨ i . s QŽ Vi . s N Ž ¨ i . s 0, hence ¨ i is nilpotent. Otherwise for every t g G m , l i Ž t . 6 s 1 and t 6 Ž ¨ i . s ¨ i for all i. This contradicts the fact that G m is a subgroup of G which acts faithfully on J X . Žii. « Ži.. Since J has non-zero nilpotents, by wAJ, Theorem 6x, the involution G is equivalent to the involution ²1, y1, 1: on J. But then SO Ž G . is isotropic and since SO Ž G . ¨ G Žsee 5.1.., G is isotropic. LEMMA 3.2. Let J be an exceptional Jordan algebra o¨ er a field k of characteristic not equal to 2 or 3. Let J X be a Jordan subalgebra of dimension G 12. Suppose that the restriction of the trace form Q of J to J X is non-degenerate. Then J X is simple and either dim k J X s 15 or J X s J. Proof. Without loss of generality we assume that k is algebraically closed. Suppose that there exists an ideal I in J X such that I 2 s 0. Let x g I. For y g J X , xy g I and hence Ž xy . 2 s 0, so that QŽ x, y . s T Ž xy . s 0. Since the trace form Q restricted to J X is assumed to be non-generate, we have x s 0. Therefore, by wJ2, Lemma, p. 239x, J X s I1 [ ??? [ In with each I j a simple ideal. Since Ii I j ; Ii l I j s 0 for i / j, the trace form Q restricted to each I j is non-degenerate. Let e j g I j be such that 1 s e1 q ??? qe n . Then each e j is an idempotent and I j is a Jordan algebra with e j as the identity element. Suppose that dim k I j G 2 for some j. Since k is algebraically closed and Q restricted to I j is non-degenerate, there exists non-trivial idempotents eXj , eYj g I j such that e j s eXj q eYj . Since T Ž e . s 1

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or 2 for any non-trivial idempotent e Žcf. wJ2, p. 359x. and T Ž1. s 3, it follows that 1 cannot be written as a sum of more than three non-trivial idempotents. Therefore either n s 1 or n s 2 and one of the two ideals is one dimensional. Suppose that n s 2 and dim k I1 s 1. Since dim k I2 G 2, e2 is a sum of two idempotents and hence e1 is a primitive idempotent in J; i.e., it cannot be written as a sum of two non-trivial idempotents in J. Let E s  x g J < e1 x s 04 . Then dim k E s 10 Žcf. wJ2, p. 367x. and I2 ; E. Since dim k I2 s dim k J X y 1 G 11, we get a contradiction. Therefore J X is simple and by wS, Theorem 2x, J X , H3 Ž C , G . for some composition algebra C . Now the lemma follows from the fact that any such algebra has dimension either 6, 9, 15, or 27.

4. EXTENSIONS OF CAYLEY AND JORDAN ALGEBRAS FROM THE AFFINE PLANE TO THE PROJECTIVE PLANE Let R be a domain in which 2 and 3 are units. Let C be a Cayley algebra over R and J an exceptional Jordan algebra over R. We call C Žresp. J . anisotropic if C mR K Žresp. J mR K . is anisotropic, where K denotes the quotient field of R. We recall that over a field k, C Žresp. J . is anisotropic if its group of automorphism Aut C Žresp. Aut J . is anisotropic as an algebraic group over k. Let n C : C ª R denote the norm form on C. Then C is anisotropic if and only if n C is anisotropic. Let T : J ª R be the trace map on the Jordan algebra J, Q : J ª R be the trace form given by QŽ x . s T Ž x 2 ., and N : J ª R be the norm form. By Ž3.1. it follows that J is anisotropic if and only if for x g J, T Ž x . s QŽ x . s N Ž x . s 0 implies that x s 0. For any integral scheme X and C a Cayley algebra bundle on X Žresp. J a Jordan algebra bundle., we say that C is anisotropic Žresp. J is anisotropic. if its restriction to the generic point is anisotropic. In this section, using methods of wKPS1x, we prove some results on extensions of Cayley and Jordan algebra bundles from the affine to the projective plane. PROPOSITION 4.1. Let R be as abo¨ e and p g R a prime element. Let J and J X be Jordan algebras o¨ er R. Suppose that J X m RrŽ p . is anisotropic. Suppose a : J m Rw1rp x ª J X m Rw1rp x is an isomorphism of Jordan algebras. Then a extends uniquely to an isomorphism a ˜ : J ª J X. Proof. It is enough to show that a Ž J . s J X . Let x g J. Suppose a Ž x . f J . Let n be the least integer such that y s p na Ž x . g J X and p ny1a Ž x . f J X . Clearly n G 1. Since a preserves the linear form T, the quadratic form Q, and the cubic form N, we have T Ž y . s p n T Ž x ., QŽ y . s p 2 n QŽ x ., and X

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N Ž y . s p 3 n N Ž x .. Hence T Ž y . s QŽ y . s N Ž y . s 0 modŽ p . and y / 0 modŽ p .. This contradicts the assumption that J X m RrŽ p . is anisotropic. PROPOSITION 4.2. Let R and p be as in Ž4.1.. Let C, CX be Cayley algebras o¨ er R. Suppose that CX m RrŽ p . is anisotropic. Suppose a : C mR Rw1rp x ª CX m Rw1rp x is an isomorphism. Then a extends to an isomorphism a ˜ : C ª CX . Proof. It is enough to show a Ž C . s CX . Let x g C with a Ž x . f CX . Let n be the least integer with y s p na Ž x . g CX and p ny1a Ž x . f CX . Clearly n / 1. Since a preserves the Cayley norm n C , we have n C X Ž y . s p 2 n n C Ž x . s 0 modŽ p .

and

y / 0 modŽ p . ,

contradicting the anisotropicity of CX m RrŽ p .. COROLLARY 4.3. Let J, J X be Jordan algebra bundles o¨ er Pkn such that a : J , J X is an isomorphism o¨ er Ank . Suppose that J and J X are anisotropic. Then a extends uniquely to an isomorphism a ˜ of J with J X o¨ er Pkn. The X same holds if J and J are replaced by Cayley algebra bundles C and C X . Proof. We give a proof for Jordan algebras which goes through verbatim for Cayley algebras in view of Ž4.2.. Let X 0 , . . . , X n be the homogeneous coordinates for Pkn. Let DŽ X n . s n A k . Let a i denote the restriction of a to DŽ X i . l DŽ X n .. Since J and J X are anisotropic, by Ž4.1., a i extends uniquely to an isometry a ˜i over DŽ X i .. Let a ˜i j denote the restriction of a˜i to DŽ X i . l DŽ X j .. We show that a ˜i j s a˜ji , so that a˜ s Ž a˜i j . is an isomorphism over Pkn. Since a˜i j and a˜ji are the restrictions of a to DŽ X i . l DŽ X j . l DŽ X n ., they coincide on this open set. By Ž4.1., a ˜i j s a˜ji . By the very construction, the extension a˜ is unique. Remark 4.4. The above corollary shows that if C Žresp. J . is an anisotropic Cayley Žresp. Jordan. algebra over A2k and C˜ Žresp. J˜. denotes its extension to Pk2 then Aut A2kC , Aut P k2 C˜ Žresp. Aut A2k J , Aut P k2 J˜.. See also Section 3. For the rest of this section we assume that char k s 0. PROPOSITION 4.5. Let H be a connected reducti¨ e algebraic group defined o¨ er k. Then e¨ ery H-bundle o¨ er A2k extends to Pk2 as an H-bundle. Proof. Let X, Y, Z be the homogeneous coordinates of Pk2 . Let x s XrZ and y s YrZ. Let E be an H-bundle over A2k s SpecŽ k w x, y x. and E0 its reduction modulo Ž x, y .. In view of wRR, Theorem 1.1x, E and E0 are isomorphic over k Ž x .w y x. Let f g k w x x be such that E , E0 over

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SpecŽ k w x, y, fy1 x.. Let f Z denote the Z homogenization of f preserving the degree of f. Let U s DŽ Z . j Ž DŽ X . l DŽ f Z ... Then the H-bundles E over DŽ Z . and E0 pulled back to DŽ X . l DŽ f Z . together with an isomorphism a : E , E0 over DŽ Z . l DŽ X . l DŽ f Z . define an H-bundle E˜ over U s Pk2 _Ž0 : 1 : 0.4 , which, in turn, extends to an H-bundle over Pk2 in view of wCTS, Theorem 6.13x. COROLLARY 4.6. E¨ ery anisotropic Jordan Ž resp. Cayley . algebra bundle on A2k admits a unique extension to Pk2 . If, further, the gi¨ en bundle on A2k admits a reduction of the structure group to a proper connected reducti¨ e subgroup of F4 or G 2 respecti¨ ely, its corresponding extension to Pk2 has the same property. Proof. This is immediate from Ž4.3. and Ž4.5.. We conclude this section with the following proposition which will be used in Section 6. PROPOSITION 4.7. Let G s Aut J 0 be anisotropic and defined o¨ er k. Let H be a connected reducti¨ e subgroup of G. Let J be a principal G-bundle o¨ er k w X, Y x such that J s J 0 mk R, bar denoting modulo Y, and R s k w Y x. Let f, g g R be such that Ž f, g . s 1. Let J m R g w Y x , J 0 m R g w Y x. Suppose J m R f w Y x has a reduction of the structure group to H. Then J has reduction of the structure group to H. Proof. The bundle J is obtained by patching J m R f w Y x over R f w Y x and J 0 over R g w Y x by an isomorphism a : J m R f g w Y x , J 0 m R f g w Y x. Since J s J 0 mk R, one may assume that a s identity. The bundle J m R f w Y x has a reduction of the structure group to H. Further in view of wRR, Theorem 1.1x, J m R f w Y x as an H-bundle, is trivial over k Ž X .w Y x. There exists h g k w X x, Ž f, h. s 1, such that J m k w X x h f w Y x is trivial as an H-bundle, i.e., there exists an isomorphism b : J m k w X x h f w Y x , J 0 m k w X x h f w Y x of H-bundles such that b s identity. We therefore have aby1 g AutŽ J 0 m k Ž X .w Y x. such that aby1 s identity. Since J 0 is anisotropic, AutŽ J 0 m k Ž X .w Y x. s AutŽ J 0 m k Ž X .. Žcf. Proposition 5.4., aby1 s identity, hence a s b over k Ž X .w Y x. Thus the trivialization a , which is an isomorphism of H-bundles generically, is an isomorphism of H-bundles over k w X x f g w Y x. Hence J has a reduction of the structure group to H. 5. AUTOMORPHISMS OF REDUCED EXCEPTIONAL JORDAN ALGEBRAS Let R be a ring in which 2 and 3 are units. Let C0 be a Cayley algebra over R. Let J 0 s H3 Ž C0 , G . denote the exceptional Jordan algebra over R defined in Section 3, corresponding to the involution given by the matrix

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G s ²g 1 , g 2 , g 3 :, g i g R*. Let e1 s ²1, 0, 0:, e 2 s ²0, 1, 0:, and e3 s ²0, 0, 1: denote the standard idempotents of H3 Ž C0 , G .. For x g C0 , let 0 X1Ž x . s 0 0

0 0 y1 g3 g2 x



0 x , 0

and X3 Ž x . s

0 

0 X2 Ž x . s 0 x

 0

x 0 0

0

gy1 2 g1 x 0

0 0 0

gy1 1 g3 x 0 0

0

0 0 . 0

We denote by X i the R-submodule of H3 Ž C0 , G . consisting of the matrices X i Ž x . and by X iX the submodule of matrices X i Ž x . with t Ž x . s 0. Let R Ž i. s  X i Ž l. ¬ l g R4 . Then X i s R Ž i. H X iX and with respect to the trace form, J 0 decomposes as J 0 s Re1 H Re2 H Re3 H X 1 H X 2 H X 3 . The trace form restricted to Re1 H Re2 H Re3 is ²1, 1, 1: and when rey1 y1 . stricted to X 1 H X 2 H X 3 it is 2Žgy1 3 g 2 n 0 H g 1 g 3 n 0 H g 2 g 1 n 0 , where n 0 s n C 0 is the norm on the Cayley algebra C0 . LEMMA 5.1. H3 Ž C0 , G ..

Let A g O Ž G .. The map u ¬ Au Ay1 is an automorphism of

Proof. Since A has entries in R, the expression Au Ay1 makes sense. For u g H3 ŽC 0 , G . we have u s Gy1u t G and y1 t

Gy1 Ž Au A

t

. GsGy1 Ž Au Ay1 . G s Gy1 Ž At

y1

u tAt . G

s Au Ay1 , since A g O Ž G . and At GA s G. It is easy to verify that this map is a Jordan algebra automorphism. We denote the above automorphism by Int A . LEMMA 5.2.

Let R be a domain. Let

a As c 0

ž

b d 0

0 0 , 1

/

a Bs 0 c

ž

0 1 0

b 0 g SO Ž G . d

/

with a, b, c, d g R such that abcd / 0. Let c 2 s Int B , c 3 s Int A . Then the following holds.

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Ži. c 2 stabilizes X 1 H X 3 and X 2X and takes e¨ ery non-zero element of X 1 outside X 1 and that of X 3 outside X 3 . Further, it fixes the idempotent of e2 and maps the rank 2 submodule R Ž2. H RŽ e1 y e3 . into itself. Žii. c 3 stabilizes X 1 H X 2 and X 3X and takes e¨ ery non-zero element of X 1 outside X 1 and that of X 2 outside X 2 . Further, it fixes the idempotent e3 and maps the rank 2 submodule R Ž3. H RŽ e1 y e2 . into itself. Proof. We give a proof of Žii.. A proof for Ži. follows on similar lines. We have 0 c3 0 0



c3



0 0 x

0 0 gy1 3 g2 x

0 0 0

0 a x s c 0 0

b d 0

s

0 0

0 0 1

0 ž /  0 ž / ycgy1 3 g2 x

gy1 a 1 g3 x s c 0 0 0 s



b d 0

0 0 1

0

0

0 dx

0 ybx

0 0 0

0 0 gy1 3 g2 x

agy1 1 g3 x

yb a 0

0 0 1

/

d yc 0

yb a 0

0 0 1

/

0

gy1 1 g3 x 0 0

0 0 0

0ž

d yc 0

bx dx , 0

0 0 agy1 3 g2 x 0 0 x

0 x 0

0ž

0

. cgy1 1 g3 x 0

Thus c 3 stabilizes X 1 H X 2 . Further, since abcd / 0, c 3 maps any nonzero element of X 1 outside X 1 and that of X 2 outside X 2 . Now,

c3



0 y1 g2 g1 x 0

b dgy1 0 2 g 1 x y acx 2 y1 s 0 d g 2 g1 x y c2 x 0 0

x 0 0

0 

2 yb 2gy1 2 g1 x q a x

yb dgy1 2

g 1 x q acx 0

0

0

0 . 0

X Ž X. Since A g SO Ž G ., b dgy1 2 g 1 q ac s 0, so that c 3 X 3 s X 3 . Finally,

c 3 Ž e1 y e 2 . s



ad q bc 2 cd 0

y2 ab y Ž bc q ad . 0

0 0 g R Ž3. H R Ž e1 y e2 . 0

0

and by the computation done before, c 3 Ž X 3 H RŽ e1 y e2 .. ; X 3 H RŽ e1 y e2 . and c 3 maps R Ž3. H RŽ e1 y e2 . into itself. Recall the Moufang identity in C0 ; a Ž xy . a s Ž ax . Ž ya . ,

a, x, y g C0 .

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For a g C0 , we define maps Ua , R a , L a of C0 into itself by Ua Ž x . s axa,

L a Ž x . s ax,

R a Ž x . s xa.

Then the Moufang identity gives Ua Ž xy . s L a Ž x . R a Ž y . . For a g C0 with n 0 Ž a. s 1, Ua , L a , and R a belong to O Ž n 0 .. Choose a g C0 _ R with n 0 Ž a. s 1 with t Ž a. / 0. We define ca : J 0 ª J 0 as

ca Ž e i . s e i , ca Ž X 3 Ž x . . s X 3 Ž L a Ž x . . ,

i s 1, 2, 3.

ca Ž X 2 Ž x . . s X 2 Ž Ua Ž x . .

and

ca Ž X 1 Ž x . . s X 1 Ž R a Ž x . . . Then ca is an automorphism of J 0 Žcf., wJ1, p. 419, Proof of Theorem 6x.. Further, for Ž x, y, z . g X 1 H X 2 H X 3 , caŽ x, y, z . s Ž xa, aya, az .. Thus ca stabilizes X 1 , X 2 and X 3 . Also since n 0 Ž a. s 1 and t Ž a. / 0, a2 f R. Hence ca maps R Ž i. into X i such that no non-zero element of R Ž i. is mapped into R Ž i. . We record this as LEMMA 5.3. The map ca : J 0 ª J 0 , defined as abo¨ e, is an automorphism of J 0 with the property that ca stabilizes each X i and maps R Ž i. into X i such that no non-zero element of R Ž i. is mapped into R Ž i. . We record here some rigidity properties of automorphisms of Jordan and Cayley algebras over polynomial rings. We compare this with a corresponding statement for orthogonal groups in wPS, Lemma 2.2x. PROPOSITION 5.4. anisotropic. Then

Let J 0 be a Jordan algebra o¨ er a domain R, which is Aut Rw Y x Ž J 0 m R w Y x . s Aut R J 0 .

Proof. Let a g Aut Rw Y xŽ J 0 m Rw Y x.. Let ¨ 0 g J 0 and a Ž ¨ 0 . s w 0 q w 1Y q ??? qwk Y k , wi g J 0 . Suppose that k G 1 and w k / 0. Then N Ž a Ž ¨ 0 .. s N Ž w k .Y 3 k q Ža polynomial of degree - 3k . s N Ž ¨ 0 . g R. Hence N Ž w k . s 0. Also, T Ž a Ž ¨ 02 .. s T Ž w k2 ..Y 2 k q Ža polynomial of degree - 2 k . s T Ž ¨ 02 . g R. Thus T Ž w k2 . s 0. Now, T Ž a Ž ¨ 0 .. s T Ž w k ..Y k q Ža polynomial of deg - k . s T Ž ¨ 0 . g R. Hence T Ž w k . s 0. So T Ž w k . s QŽ w k . s N Ž w k . s 0. Since J 0 is anisotropic, w k s 0. This leads to a contradiction. Thus a Ž ¨ 0 . g J 0 and a s a 0 m 1 for a 0 g Aut R J 0 . This proves the proposition.

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PROPOSITION 5.4X . Let C0 be an anisotropic Cayley algebra o¨ er R. Then Aut Rw Y xŽ C0 m Rw Y x. s Aut R C0 . Proof. This is immediate from wPS, Lemma 2.2x since n C 0 is anisotropic and automorphisms preserve norm. PROPOSITION 5.5. Let R be a domain and p, h g R with Rp q Rh s R. Let J1 , J 2 be anisotropic Jordan algebras o¨ er Rw Y x which are extended after in¨ erting p. Let w : Ž J1 . h ª Ž J 2 . h be an isometry o¨ er R h w Y x such that w is defined o¨ er R. Then w is defined o¨ er Rw Y x. Proof. The proof goes on parallel lines as that of wP, 2.4x, using Ž5.4..

6. CONSTRUCTION OF NON-TRIVIAL JORDAN ALGEBRAS OVER k w X, Y x We begin with the following ‘‘general nonsense’’ lemma which will be used in the sequel. LEMMA 6.1. Let R be a domain and A an algebra Ž not necessarily associati¨ e . o¨ er Rw Y x. Let A 0 be an algebra o¨ er R such that A, as an Rw Y x module, is isomorphic to A 0 m Rw Y x and A is isomorphic to A 0 as an R-algebra, bar denoting modulo Y. Then there exists an algebra structure AX on A 0 mR Rw Y x, isomorphic to A, such that AXs A 0 . Ž Here, the natural map A 0 mR R w Y x , A 0 is treated as an identification.. Proof. Let b : A 0 ª A be an isomorphism of algebras over R. Let a : A 0 mR Rw Y x , A be an isomorphism of Rw Y x-modules and a : A 0 ª A its reduction modulo Y. Let b s a (ŽŽ ay1 ( b . m 1.. We pull back the algebra structure on A to an algebra structure on AX s A 0 m Rw Y x through b : A 0 m Rw Y x ª A, i.e., we set, for x, y g AX x ) y s by1 Ž b Ž x . . b Ž y . . . Then b : AX ª A is an isomorphism of Ž AX , ). with Ž A, .. Further, x ) y s by1 Ž b Ž x . . b Ž y . . s by1 Ž b Ž x. y . . s x ( y, where x ( y denotes multiplication in A 0 . Thus AX s A 0 . Let k be a field of characteristic not equal to 2 and 3, admitting a reduced anisotropic Jordan algebra J 0 s H3 Ž C0 , G ., where C0 is a Cayley division algebra over k and G s ²g 1 , g 2 , g 3 :, g i g k*. Let G s Aut J 0 . In this section, we construct, in several steps, a family of Jordan algebras J w ix

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over k w X, Y x which are rigid in the following sense: the principal G-bundle associated to J w ix over A2k does not admit reduction of the structure group to any proper connected reductive subgroup of G. This will be shown in Section 7. We begin with the construction. Step I. By wT, Theorem 3.1x, there exist infinitely many pairs Ž Ci , pi . i G 1 of mutually non-isomorphic Cayley algebras Ci over k w X, Y x and polynomials pi g k w X x with Ž pi , pj . s 1 for i / j, with the following properties: if bar denotes reduction modulo Y, Ci , C0 m k w Y x, Ci m k w X x p i w Y x , C0 m k w X x p i w Y x and the rank 7-quadratic space CiX of trace zero elements of Ci is indecomposable over k w X, Y x. Let nX denote the norm on the Cayley algebra restricted to trace zero elements. Twisting Ci through an isomorphism bi : Ci , C0 m k w Y x, as in Ž6.1., we may and do assume that Ci s C0 m k w Y x. Let p i : Ci m k w X x p i w Y x , C0 m k w X x p i w Y x be isomorphism such that p i s identity. Let Ji s H3 Ž Ci , G ., i G 1. Then Ji s J 0 s H3 Ž C0 , G . and p i gives rise to the isomorphism

p i : Ji m k w X x p i w Y x , J 0 m k w X x p i w Y x of Jordan algebras with p i s identity. PROPOSITION 6.2. The algebras Ji are mutually non-isomorphic. Proof. Suppose Ji , J j for i / j. Since Ji is extended from J 0 over k w X x p i w Y x and J j from J 0 over k w X x p j w Y x with Ž pi , pj . s 1, Ji on k w X xw Y x, is locally extended from k w X x and by wBCW, 4.15x, Ji , J 0 m k w X, Y x. Let C˜i be an extension of the Cayley algebra Ci to Pk2 . Then J˜i s H3 Ž C˜i , G . is the extension of Ji to Pk2 , by the unique extension property Ž4.6.. The underlying vector bundle of J˜i decomposes as O e1 [ O e2 [ O e3 [ Ž X˜1X [ O . [ Ž X˜2X [ O . [ Ž X˜3X [ O . , where O denotes the structure sheaf of Pk2 and X˜jX the trace zero elements in C˜i sitting in 3 copies of C˜i in J˜i . The bundles X˜jX are absolutely indecomposable ŽProposition 2.3, Corollary 2.4.. Hence J˜i is non-trivial as a vector bundle. On the other hand, Ji , J 0 m k w X, Y x and p *J 0 , J˜i , where p : Pk2 ª Spec k is the structure morphism, by the uniqueness of the extension. However, p *J 0 is trivial as a vector bundle, leading to a contradiction. This proves the proposition. Step II. In this step, we patch the bundles Ji locally to get new bundles on A2k . More precisely, we take the algebra Ž J 2 iy1 . p 2 i over k w X x p 2 i w Y x and Ž J2 i . p over k w X x p 2 i w Y x and patch them over k w X x p 2 iy 1 p 2 i w Y x through an 2 iy 1 isomorphism

u 1Ž i. : Ž J2 iy1 . p 2 iy 1 p 2 i , Ž J 2 i . p 2 iy 1 p 2 i defined as follows.

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Let a g C0 _ k be an element with n 0 Ž a. s 1 and t Ž a. / 0. Let ca : J 0 ª J 0 be the automorphism defined in Ž5.3.. We set u 1Ž i. s py1 2 i ( ca ( p 2 iy1 . Let J i denote the resulting Jordan algebra. Let qi s p 2 iy1 p 2 i . Then Ž qi , q j . s 1 for i / j and by the very construction, Ž J i .q , J 0 m k w X x q w Y x. i i Writing Ji s H3 Ž Ci , G . as Re1 [ Re2 [ Re3 [ X 1, i [ X 2, i [ X 3, i Žcf. Section 5., with R s k w X, Y x, the map u 1Ž i. patches the rank 8 submodule X j, 2 iy1 of J 2 iy1 with X j, 2 i of J 2 i to yield a rank 8-submodule X ji of J i for j s 1, 2, 3. Further, it patches the idempotents e j of J 2 iy1 with the corresponding e j of J 2 i . Thus J i decomposes as J i s Re1 [ Re2 [ Re3 [ X 1i [ X 2i [ X 3i . PROPOSITION 6.3. The quadratic spaces X ji, for the trace form, are rigid. Further, X ji is not similar to X ki for j / k, j, k s 1, 2, 3. Proof. The space X ji is obtained by patching the quadratic space l j n C 2 iy 1 s l j Ž1 H nXC 2 iy 1 . on k w X x p 2 i w Y x and n C 2 i s l j Ž1 H nXC 2 i . on y1 y1 k w X x p 2 iy 1w Y x by isometries py1 2 i ( R a ( p 2 iy1 , p 2 i (Ua ( p 2 iy1 and p 2 i ( L a (p 2 iy1 , respectively, with l j g k* for j s 1, 2, 3. The rank 7 spaces nXC j are indecomposable over k w X, Y x and hence rigid Ž2.4.. Further, since t Ž a. / 0, R a , Ua , L a : C0 ª C0 map 1 away from 1. Hence by Ž2.6., the rank 8 space X ji is rigid for j s 1, 2, 3. We now show that X ji is not similar to X ki for j / k. We do this for j s 1, k s 2. Other cases follow on similar lines. Suppose X 1i , l X 2i for some l g k*. Then there exist isometries w 2 i : 1 H nXc 2 iy 1 ª lŽ1 H nXc 2 iy 1 . over k w X x p 2 i w Y x and w 2 iy1 : 1 H nXC 2 i ª lŽ1 H nXC 2 i . over k w X x p 2 iy 1w Y x, such that the diagram

lŽ1 H nXc 2 iy 1 .

py1 2 i ( R a ( p 2 iy1

6

6

1 H nXC 2 i

w 2 iy1

p y1 2 i (Ua ( p 2 iy1

6

w2i

6

1 H nXc 2 iy 1

lŽ1 H nXC 2 i .

is commutative. We have, by reducing modulo Y, Ua ( w 2 i s w 2 iy1 ( R a . Since Ua and R a are defined over k and Ž p 2 i , p 2 iy1 . s 1, it follows wP, 2.3x that w 2 i and w 2 iy1 both extend to isometries of the corresponding spaces over k w X, Y x. Since nXC 2 iy 1 and nXC 2 i are both rigid, in view of Ž2.5., w 2 i and w 2 iy1 break up into isometries of ²1: , ² l: and nXC 2 iy 1 , l nXC 2 iy 1 and nXC 2 i , l nXC 2 i , respectively. It follows that l is a square and, after scaling, we may assume l s 1. For g q x g C2 i s R.1 H C2X i we have

w 2 i Ž g q x . s e1g q e 2 x

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and for g q x g C2 iy1 s R.1 H C2 iy1

w 2 iy1 Ž g q x . s e 3 g q e4 x with e i s "1 for 1 F i F 4. Commmutativity of the above diagram modulo Y shows that if a s g 0 q x 0 , Ua ( w 2 i Ž1. s "a2 , w 2 iy1 ( R aŽ1. s e 3 g 0 q e 4 x 0 s "a2 . This contradicts the choice of a such that t Ž a. / 0 and n C 0Ž a. s 1. This proves the proposition. PROPOSITION 6.4. The algebras J i, J j are non-isomorphic for i / j. Proof. Suppose J i , J j for i / j. Then J i would be extended from J 0 after inverting qi and q j , which are coprime. In vew of wBCW, 4.15x, J i , J 0 m k w X , Y x. This implies that J 2 i m k w X x p 2 iy 1w Y x , J i m k w X x p 2 iy 1w Y x , J 0 m k w X x p 2 iy 1w Y x. However, J 2 i m k w X x p 2 i w Y x , J 0 m k w X x p 2 i w Y x, so that by wBCW, 4.15x, J 2 i , J 0 m k w X, Y x, contradicting Ž6.2.. This proves the proposition. We replace J i by a suitable twist Žcf. Lemma 6.1., and assume that J s J0 . i

Step III. We take Ž J 2 iy1 .q 2 i over k w X x q 2 i w Y x and Ž J 2 i .q 2 iy 1 over k w X x q 2 iy 1w Y x and patch them over k w X x q 2 iy 1 q 2 i w Y x by an isomorphism

u 2Ž i. : Ž J 2 iy1 . q 2 iy 1 q 2 i ª Ž J 2 i . q 2 iy 1 q 2 i defined as follows. Let p i : J i m k w X x q i w Y x , J 0 m k w X x q i w Y x be an isomorphism such that i p s identity. Let c 3 : J 0 ª J 0 be the automorphism defined in Ž5.2.. We set u 2Ž i. s Žp 2 i .y1 ( c 3 (Žp 2 iy1 .. Let J Ž i. denote the resulting Jordan algebra. In a similar fashion, we construct the Jordan algebra J i4 through the patching u 3Ž i. which replaces u 2Ž i. above, defined by

u 3Ž i. s Ž p 2 i .

y1

( c 2 ( Ž p 2 iy1 . .

Let ri s q2 iy1 q2 i . Then Ž ri , r j . s 1 for i / j. By the construction, Ž J Ž i. . r i , J 0 m k w X x r i w Y x. The map u 2Ž i. patches the rank 16 subspace X 12 iy1 H X 22 iy1 of J 2 iy1 over k w X x q 2 i w Y x and X 12 i H X 22 i of J 2 i over k w X x q 2 iy 1w Y x to yield a subspace X 12i of rank 16 in J Ž i.. Further, it patches the subspace X 32 iy1 H RŽ e1 y e2 . of J 2 iy1 with X 32 i H RŽ e1 y e2 . of J 2 i to yield a subspace i Ž i. XŽ3, of rank 9. Since c 3 fixes e1 q e2 y 2 e3 , J Ž i. has a rank 1 1r2. of J trivial summand generated by e1 q e2 y 2 e3 . Thus J Ž i. decomposes as an orthogonal bundle as J Ž i. s X 12i H XŽ3i , 1r2. H R Ž e1 q e2 y 2 e3 . H R Ž e1 q e2 q e3 . .

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The bundle J i4 has a similar decomposition J i4 s X 13i H XŽ2i , 1r3. H R Ž e1 q e3 y 2 e2 . H R Ž e1 q e2 q e3 . . i Ž i. Ž i PROPOSITION 6.5. The subspaces X 12i and XŽ3, resp. X 13 and 1r2. of J i4 of J are rigid and hence indecomposable.

i XŽ2, 1r3.

i Proof. The space X 12 is obtained by patching the sum X 12 iy1 H X 22 iy1 2i of rigid subspaces of J through u 2Ž i. with u 2Ž i. s c 3 . The map c 3 maps every non-zero element of X 1 outside X 1 and that of X 2 outside X 2 . Further, X 1j is not similar to X 2j for any j Ž6.3.. In view of Ž2.6., it follows that X 12i is rigid. Since u 2Ž i. patches the subspace X 32 iy1 H RŽ e1 y e2 . of J 2 iy1 with X 32 i H RŽ e1 y e2 . of J 2 i with u 2Ž i. s c 3 mapping k Ž e1 y e2 . i outside k Ž e1 y e2 . and c 3 Ž X 3 . o X 3 , in view of Ž2.6., XŽ3, 1r2. is rigid. A i4 similar argument yields the proposition for J as well.

PROPOSITION 6.6. The algebras J Ž i., J Ž j. are non-isomorphic for i / j. Similarly, the algebras J i4 and J i4 are non-isomorphic for i / j. Proof. Suppose J Ž i. , J Ž j., i / j. Arguing as in the proof of Ž6.4., J Ž i. is isomorphic to J 0 over k w X x r i w Y x as well as k w X x r j w Y x and hence is isomorphic to J 0 m k w X, Y x wBCW, 4.15x. Hence J 2 iy1 m k w X x q 2 i w Y x , J Ž i. mk w X x q 2 i w Y x , J 0 m k w X x q 2 i w Y x. Further, J 2 iy1 m k w X x q 2 iy 1w Y x , J 0 m k w X x q 2 iy 1w Y x and this would imply that J 2 iy1 , J 0 m k w X, Y x, contradicting Ž6.4.. That the algebras J i4, J  j4 are non-isomorphic for i / j follows on similar lines. We replace J Ž i. and J i4 by a suitable twist Ž6.1. and assume, without loss of generality, that J Ž i.s J 0 s J i4. Step IV. We take the space JrŽiqi. 1 over k w X x r iq 1w Y x and Jriq14 over i k w X x r i w Y x and patch them over k w X x r i r iq 1w Y x by an isometry

u4Ž i. : J Ž i. m k w X x r i r iq 1 w Y x ª J iq14 m k w X x r i r iq 1 w Y x defined as follows. Let

p Ž i. : J Ž i. m k w X x r i w Y x ª J 0 m k w X x r i w Y x pi4 : J i4 m k w X x r i w Y x ª J 0 m k w X x r i w Y x be isomorphisms with p Ž i. s identity, pi4 s identity. We set u4Ž i. s w ix py1 be the resulting Jordan algebra over k w X, Y x. Let iq14p Ž i. . Let J si s ri riq1. Then Ž si , s j . s 1 for i / j and by construction, Ž j w ix .s i , J 0 m K w X x s i w Y x.

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PROPOSITION 6.7. The algebras J w ix and J w j x are non-isomorphic for i / j. Proof. The proof runs on exactly the same lines as Ž6.4. noting that J w ix m k w X x s i w Y x , J 0 m k w X x s i w Y x for all i and using Ž6.5.. We shall show in the next section that the algebras J w ix are rigid in the X w ix following sense. If G s Aut J 0 and J denotes the trace zero elements in X J w ix, J w i x as a G-bundle, admits no reduction of the structure group to any proper connected reductive subgroup of G.

7. F4 BUNDLES WITH NO REDUCTION OF THE STRUCTURE GROUP TO PROPER CONNECTED REDUCTIVE SUBGROUPS In this section we prove that the principal G-bundles associated to the Jordan algebras J w ix over A2k , constructed in Section 6, have no reduction of the structure group to any proper connected reductive subgroup. We begin with some lemmas on representations of algebraic groups. LEMMA 7.1. Let G be an algebraic group o¨ er a field k. Let V be a representation of G o¨ er k. Suppose that V m k , ŽW [ ??? [ W . mk k for some representation W of G which is irreducible o¨ er the algebraic closure k of k. Then V , W [ ??? [ W. Proof. Since V mk k , ŽW [ ??? [ W . mk k, it corresponds to an element in H 1 Ž k, Aut G ŽŽW [ ??? [ W . mk k ... Since W mk k is an irreducible representation of G, we have H 1 Ž k, Aut G ŽŽW [ ??? [ W . mk k .. , H 1 Ž k, GLn . s 0. Therefore V , W [ ??? [ W. We, now on, assume that char k s 0. LEMMA 7.2. Let H be a simply connected, simple algebraic group o¨ er k of type A1. Let V0 be the 2 dimensional representation of H o¨ er k. Then the symmetric power Sym2 n Ž V0 . of V0 is defined o¨ er k for all n. Proof. If H is isotropic over k then V0 is defined over k. So assume H is anisotropic over k. Then H , SLŽ D . for some quaternion division algebra D over k and Sym2 Ž V0 . , D 0 m k where D 0 is the space of trace zero elements of D. Thus Sym2 Ž V0 . is defined over k. We prove the lemma by induction on n. Assume that Sym2 i Ž V0 . is defined over k for 1 F i F n y 1. We have Sym2Ž ny1. Ž V0 . m Sym2 Ž V0 . , Sym2 n Ž V0 . [ Sym2Ž ny1. Ž V0 . [ ??? [ Sym0 Ž V0 .

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Žcf. wOV, p. 300x.. By induction, Sym2 i Ž V0 . is defined over k for 1 F i F n y 1, thus Sym2Ž ny1. Ž V0 . m Sym2 Ž V0 . is defined over k and hence Sym2 n Ž V0 . is defined over k. LEMMA 7.3. Let H be as in Ž7.2.. Suppose further that H is anisotropic. Let V be a faithful representation of H of dimension 2 p, with p an odd prime. If V m k is reducible, then V is reducible. Proof. Suppose that V m k is reducible and V is irreducible. Then V m k , W [ W, W being the p-dimensional irreducible representation of H or V m k , V0 [ ??? [ V0 , V0 being the 2-dimensional irreducible representation of H. Since W , Sym py 1 Ž V0 ., p being odd, it is defined over k Ž7.2.. Thus if V m k , W [ W, then by Ž7.1., V is reducible, contradicting the assumption. Suppose that V m k , V0 [ ??? [ V0 . Since V0 is defined over a quadratic extension L of k by Ž7.1., V mk L , V0 [ ??? [ V0 . We note that the non-trivial automorphism s of L over k permutes the V0 summands. Since p is odd and there are p copies of V0 in V mk L, it follows that there is a summand V0 which is s-stable, contradicting the assumption on H. This completes the proof. LEMMA 7.4. Let H be a semisimple algebraic group o¨ er an algebraically closed field and V a finite dimensional representation of H. Suppose that there is a non-singular quadratic form q on V which is H-in¨ ariant. If V s W1 [ W2 , where W2 is the isotypical component for the tri¨ ial representation and W1 is H-stable, then the restriction of q to W1 is non-degenerate. Proof. Let b be the bilinear form associated to q. Let U s  x g W1 ¬ Ž b x, y . s 0 for all y g W14 be the radical of q. Since q is H-invariant, H stabilizes U. Suppose that U / 0. Let U X be an irreducible subrepresentation of U. By the assumption on W1 , dim k U X G 2. Let H X be a simple factor of H such that H X acts non-trivially on U X . Let T be a maximal torus of H X . Since k is algebraically closed, the action of T on U X is diagonalisable. Let x g U X be an eigenvector with eigencharacter x for T. Since q is non-singular on V, there exists y g V such that bŽ x, y . s 1. Since bŽ x, w . s 0 for all w g W1 , we assume that y g W2 . For any t g T, we have 1 s b Ž x, y . s b Ž tx, ty . s b Ž x Ž t . x, y . s x Ž t . b Ž x, y . s x Ž t . . Thus the action of T on U X is trivial and hence H X acts trivially on U X , a contradiction. Let G be a connected reductive group over k and V a finite dimensional representation of G. We say that V is of type n if V is a sum of irreducible representations of dimension n. Suppose that V is irreducible. Then it is easily verified that Vk s V mk k is of type n, for some n.

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Let C0 be a Cayley division algebra over k. Let G s ²g 1 , g 2 , g 2 :, g i g k* and J 0 s H3 Ž C0 , G .. Let G s Aut J0 . Let V be the subspace of trace zero elements in J 0 . LEMMA 7.5. Let H be a proper connected reducti¨ e subgroup of G. Then H acts reducibly on V. Proof. Replacing H by the commutator subgroup of H, we assume that H is semisimple. Suppose that the action of H on V is irreducible. Then Vk as a H-representation is of type n, for some n. Since the action of Hk s H = k on Vk is reducible wD, Theorem 16.1, p. 239x and dim k Ž V . s 26, for the action of Hk , Vk decomposes as [Wi with Wi irreducible and dim kWi s 13, i s 1, 2 or dim k ŽWi . s 2, 1 F i F 13. Let H1 be a simple factor of Hk . Since the action of H on Vk is faithful, there exists an i such that H1 acts faithfully on Wi . Since dim kWi s 2 or 13, it follows, from the table of formulae for dimensions of irreducible representations of simple groups over an algebraically closed field wOV, pp. 300]305x, that H is of type A1. By Ž7.3., Hk has more than one simple component. We write Hk as an almost direct product H1 H2 , with H1 simple. We observe that every irreducible representation of H1 H2 decomposes as V1 m V2 , where Vi is an irreducible Hi-representation, for i s 1, 2. Since each Wi is an irreducible Hk-representation of dimension a prime, the action of H1 on each Wi is either irreducible or trivial. Further, if the action of H1 is irreducible on Wi for some i, then the action of H2 on Wi is trivial. Since the action of H on V is faithful, each Hi acts non-trivially on some Wj . Let us first consider the case where Vk s W1 [ W2 , with dim kWi s 13, i s 1, 2. Then H2 is also simple. Without loss of generality we assume that H1 acts irreducibly on W1. Then H2 acts trivially on W1 and irreducibly on W2 . Therefore, by Ž7.4., the restriction of the trace form of J to W1 is non-degenerate. Let J1 be the Jordan subalgebra of J generated by W1. Since H2 acts trivially on W1 , it acts trivially on J1. Suppose that k [ W1 is a proper subspace of J1. Then dim k J1 G 15 and J1 l W2 / 0 is a trivial subrepresentation of W2 for H2 , leading to a contradiction. Hence J1 s k [ W1. Since dim k J1 s 14 and the trace form restricted to k [ W1 is non-degenerate, this once again leads to a contradiction in view of Ž3.2.. We may therefore assume that Vk s W1 [ ??? [ W13 , with dim kWi s 2, 1 F i F 13. Let r, 1 F r F 13, be such that H1 acts irreducibly on Wi for 1 F i F r and trivially on Wj for r q 1 F j F 13. Then H2 acts trivially on Wi for 1 F i F r. By Ž7.4., the trace form restricted to Wrq1 [ ??? [ W13 is non-degenerate. Since H has at least two simple factors, without loss of generality we assume that r F 6. Let J 2 be the Jordan subalgebra of J generated by Wrq1 , . . . , W13 . Then H1 acts trivially on J 2 . Since r F 6, we have dim k J 2 G 14. Arguing as above and using Ž3.2., Ž7.4., we get ŽW1

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[ ??? [ Wr . l J 2 / 0. Since H1 acts irreducibly on Wi for 1 F i F r and trivially on J 2 , this is a contradiction. This proves the lemma. PROPOSITION 7.6. The Jordan algebra J Ž i. Ž resp. J i4 . admits an extension J˜ Ž resp. J˜i4 . to the projecti¨ e plane Pk2 whose underlying bundle has a decomposition Ž i.

J˜Ž i. s X˜12i H X˜Ž3i , 1r2. H O Ž e1 q e2 y 2 e3 . H O Ž e1 q e2 q e3 . , i i where X˜12i extends the orthogonal bundle X 12 , X˜Ž3, 1r2. extends the orthogonal i i i bundle XŽ3, 1r2. . Further, the bundles X˜12 and X˜Ž3, 1r2. are absolutely indecomposable. A similar claim holds for J i4 with

J˜i4 s X˜13i H X˜Ž2i , 1r3. H O Ž e1 q e3 y 2 e2 . H O Ž e1 q e2 q e3 . . Proof. The bundle J Ž i. admits reduction of the structure group G to the connected reductive subgroup H which fixes the idempotent e3 Žin fact H , Spin 9 .. Thus by Ž4.6., J Ž i. admits a unique extension J˜Ž i. to Pk2 which has a reduction of the structure group to H. Since H maps X 1 H X 2 into itself, X 3 H k Ž e1 y e2 . into itself and fixes e1 q e2 y 2 e3 , J˜Ž i. has a i i decomposition as given in the proposition. Further, since X 12 and XŽ3, 1r2. i are rigid Ž6.5., it follows that X˜12 and X˜Ž3, 1r2. are absolutely indecomposable. Proof of the claims for J i4 follows on similar lines. An exceptional Jordan algebra J over A2k , whose specialization at a rational point is J, gives rise to an etale 1-cocycle EJ with values in Aut J s G. To EJ is associated a principal G-bundle PJ on A2k . Let V be the space of trace zero elements in J. For the faithful representation G ¨ Aut V s GL26 , the vector bundle associated to PJ is simply the sub-bundle J0 of trace zero elements of J. PROPOSITION 7.7. Let J w ix be the Jordan algebra constructed in Section 6. Then, the principal G-bundle PJ w i x admits no reduction of the structure group to any connected reducti¨ e subgroup H ; Ž GLŽ V1 . = GLŽ V2 .. l G, where Vi are non-zero subspaces of V with V s V1 [ V2 . Proof. Suppose the G-bundle PJ w i x has a reduction of the structure group to a connected reductive subgroup H ; Ž GLŽ V1 . = GLŽ V2 .. l G. Then ŽŽ J w ix . 0 . r iq 1 , ŽŽ J Ž i. . 0 . r iq 1 over k w X x r iq 1w Y x has a reduction of the structure group to H. In view of Ž4.7., PJ Ž i. over k w X, Y x itself has a reduction of the structure group to H. Let J˜Ž i. denote the extension of the Jordan algebra J Ž i. to Pk2 . The J˜Ž i. also admits a reduction of the structure group to H Ž4.6. and hence to GLŽ V1 . = GLŽ V2 .. Hence Ž J˜Ž i. . 0 decomposes as E1 [ E2 with Ej a GLŽ Vj .-bundle, j s 1, 2. We assume without

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loss of generality that dim V1 F 13. By the uniqueness of extension Ž4.6. and in view of Ž7.6.,

Ž J˜Ž i. . 0 s X˜12i H X˜Ž3i , 1r2. H O Ž e1 q e2 y 2 e3 . with indecomposable summands of ranks 16, 9, and 1 respectively. The i Ž only direct summands of Ž J˜Ž i. . 0 of dimension F 13 are X˜Ž3, 1r2. , O e1 q e2 y 2 e3 . and X˜Ž3, 1r2. H O Ž e1 q e2 y 2 e3 ., which have rank 9, 1, and 10, respectively. We thus have E1 s X˜Ž3i , 1r2.

if dim V1 s 9

s O Ž e1 q e 2 y 2 e 3 .

if dim V1 s 1

i s X˜Ž3, 1r2. H O Ž e1 q e 2 y 2 e 3 .

if dim V1 s 10.

A similar consideration shows that

Ž J˜iq14 . 0 s F1 [ F2

with Fj a GL Ž Vj . -bundle

with F1 s XŽ2i , 1r3.

if dim V1 s 9

s O Ž e1 q e 3 y 2 e 2 .

if dim V1 s 1

i s X˜Ž2, 1r3. H O Ž e1 q e 3 y 2 e 2 .

if dim V1 s 10.

Since J Ž i. ' J 0 ' J iq14 modŽ X, Y ., we have E1 ' V1 ' F1 modŽ X, Y .. Since XŽ3, 1r2. ' X 2 H k Ž e1 y e2 . modŽ X, Y . and XŽ2, 1r3. ' X 2 H k Ž e1 y e3 . modŽ X, Y ., we have the following Ži. k Ž e1 q e2 y 2 e3 . s k Ž e1 q e3 y 2 e2 . if dim V1 s 1 Žii. X 3 H k Ž e1 y e2 . s X 2 H k Ž e1 y e3 . if dim V1 s 9 Žiii. X 3 H k Ž e1 y e2 . H k Ž e1 q e2 y 2 e3 . s X 2 H k Ž e1 y e3 . H k Ž e1 q e3 y 2 e2 . if dim V1 s 10. Each of the above equalities clearly leads to a contradiction. This proves the proposition. THEOREM 7.8. The Jordan algebras J w ix, constructed in Section 6, gi¨ e rise to G s Aut J-bundles PJ w i x which are mutually non-isomorphic and specialize at any point of A2k to PJ . Further, they do not admit reduction of the structure group to any proper connected reducti¨ e subgroup H of G. Proof. We only need to show that Ž J w i x . 0 has no reduction of the structure group to any proper connected reductive subgroup H of G.

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Suppose it admits such a reduction. Since H acts reducibly on the space V of trace zero elements of J, Ž7.5., V s V1 [ V2 with dim k Vi G 1 and each Vi is H-invariant. Hence H ¨ Ž GLŽ V1 . = GLŽ V2 .. l G. This contradicts Ž7.7. and completes the proof. Remark 7.9. Suppose that the trace form Q restricted to the trace zero elements of J is anisotropic. This is the case, for example, if J is an anisotropic Jordan algebra over a real closed field or a number field. Then the Jordan algebra J w i x constructed in Section 6 has the further property that the trace form restricted to the 26 dimensional sub-space of trace zero elements is rigid. This can be proved by using a subtler variant of Ž2.6..

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A. A. Albert and N. Jacobson, On reduced exceptional simple Jordan algebras, Ann. Math. 66, No. 3 Ž1957., 400]417. wBCWx H. Bass, E. H. Connel, and D. L. Wright, Locally polynomial algebras are symmetric algebras, In¨ ent. Math. 38 Ž1977., 279]299. wBox A. Borel, ‘‘Linear Algebraic Groups,’’ Proc. Sympos. Pure Math., Vol. 9, pp. 3]19, Amer. Math. Soc., Providence, 1966. wCTSx J.-L. Colliot-Thelene ´ ` and J.-J. Sansuc, Fibres ´ quadratiques et composantes connexes reelles, Math. Ann. 244 Ž1979., 105]134. ´ wDx E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, in Amer. Math. Soc. Transl. Ser. 2, Vol. 6, pp. 111]244, Amer. Math. Soc., Providence, 1957. wJ1x N. Jacobson, Some groups of transformations defined by Jordan algebras, II, J. Reine Angew. Math. 204 Ž1960., 74]98. wJ2x N. Jacobson, Structure and representations of Jordan algebras, in Amer. Math. Soc. Colloq. Publ., Vol. 39, Amer. Math. Soc., Providence, 1968. wKPS1x M. A. Knus, R. Parimala, and R. Sridharan, Non free projective modules over Hw X, Y x and stable bundles over P 2 ŽC., In¨ ent. Math. 65 Ž1981., 13]27. wKPS2x M. A. Knus, R. Parimala, and R. Sridharan, On compositions and triality, J. Reine Angew. Math. 457 Ž1994., 45]70. wOVx A. L. Onishchik and E. B. Vinberg, Lie groups and algebraic groups, in Springer Series in Soviet Mathematics, Springer-Verlag, New YorkrBerlin, 1990. wPx R. Parimala, Indecomposable quadratic spaces over the affine plane, Ad¨ . Math. 62 Ž1986., 1]6. wPSx R. Parimala and R. Sridharan, A local global principle for quadratic forms over polynomial rings, J. Algebra 74 Ž1982., 264]269. wQSSx H.-G. Quebbemann, W. Scharlau, and M. Schulte, Quadratic and hermitian forms in additive and Abelian categories, J. Algebra 59 Ž1979., 264]289. wRx M. S. Raghunathan, Principal bundles on affine space and bundles on the projective line, Math. Ann. 285 Ž1989., 309]332. wRRx M. S. Raghunathan and A. Ramanathan, Principal bundles on the affine line, Proc. Indian Acad. Sci. 93 Ž1984., 137]145. wSx T. A. Springer, On a class of Jordan algebras, Nederl. Akad. Wetensch. Proc. Ser. A 62 Ž1959., 254]264. wTx Maneesh L. Thakur, Cayley algebra bundles on A2K revisited, Comm. Algebra 23, No. 13 Ž1995., 5119]5130.