where C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f ).

Classi¬cation of simple groups of type F4 . Consider the groupoid F4 =

F4 (F ) of exceptional Jordan algebras of dimension 27 over F (see §?? below if

char F = 2 and §?? if char F = 2), where morphisms are F -algebra isomorphisms.

Denote by F 4 = F 4 (F ) the groupoid of simple groups of type F4 over F , where

morphisms are group isomorphisms. By Theorem (??) we have a functor

S : F4 (F ) ’ F 4 (F ), S(J) = Autalg (J).

(26.18) Theorem. The functor S : F4 (F ) ’ F 4 (F ) is an equivalence of cate-

gories.

Proof : Let “ = Gal(Fsep /F ). The ¬eld extension functor j : F4 (F ) ’ F4 (Fsep )

is clearly a “-embedding. We ¬rst show that the functor j satis¬es the descent

condition. Let J be some object in F4 (F ) (a split one, for example). If char F = 2,

consider the F -vector space

W = HomF (J —F J, J),

the multiplication element w ∈ W and the natural representation

ρ : GL(J) ’ GL(W ).

By Proposition (??) the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. We have the functor

T = T(F ) : A(ρsep , w) ’ F4 (F )

taking w ∈ A(ρsep , w) to the F -vector space J with the Jordan algebra structure

de¬ned by w . A morphism from w to w is an element of GL(J) and it de¬nes an

isomorphism of the corresponding Jordan algebra structures on J. The functor T

has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ F4 (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

For the proof of the theorem it su¬ces by Proposition (??) to show that for any

J ∈ F4 (F ) the functor S for a separably closed ¬eld F induces a group isomorphism

Autalg (J) ’ Aut Autalg (J) .

This follows from the fact that automorphisms of simple groups of type F4 are inner

(Theorem (??)).

If char F = 2, an exceptional Jordan algebra of dimension 27 is (see §?? below)

a datum (J, N, #, T, 1) consisting of a space J of dimension 27, a cubic form N : J ’

F , the adjoint # : J ’ J of N , which is a quadratic map, a bilinear trace form T ,

and a distinguished element 1, satisfying certain properties (given in §??). In this

case we consider the F -vector space

W = S 3 (J — ) • S 2 (J — ) — J • S 2 (J — ) • F

and complete the argument as in the preceding cases.

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 373

Classi¬cation of simple groups of type G2 . Consider the groupoid G2 =

G2 (F ) of Cayley algebras over F , where morphisms are F -algebra isomorphisms.

Denote by G 2 = G 2 (F ) the groupoid of simple groups of type G2 over F , where

morphisms are group isomorphisms. By Theorem (??) there is a functor

S : G2 (F ) ’ G 2 (F ), S(C) = Autalg (C).

(26.19) Theorem. The functor S : G2 (F ) ’ G 2 (F ) is an equivalence of cate-

gories.

Proof : Let “ = Gal(Fsep /F ). The ¬eld extension functor j : G2 (F ) ’ G2 (Fsep )

is clearly a “-embedding. We ¬rst show that the functor j satis¬es the descent

condition. Let C be some object in G2 (F ) (a split one, for example). Consider

the F -vector space W = HomF (C —F C, C), the multiplication element w ∈ W ,

and the natural representation ρ : GL(C) ’ GL(W ). By Proposition (??) the

“-embedding i : A(ρsep , w) ’ A(ρsep , w) satis¬es the descent condition. We have a

functor

T = T(F ) : A(ρsep , w) ’ G2 (F )

which takes w ∈ A(ρsep , w) to the F -vector space C with the Cayley algebra

structure de¬ned by w . A morphism from w to w is an element of GL(C) and it

de¬nes an isomorphism between the corresponding Cayley algebra structures on C.

The functor T has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ G2 (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

For the proof of the theorem it su¬ces by Proposition (??) to show that for any

J ∈ G2 (F ) the functor S for a separably closed ¬eld F induces a group isomorphism

Autalg (C) ’ Aut Autalg (C) .

This follows from the fact that automorphisms of simple groups of type G2 are

inner (Theorem (??)).

26.B. Algebraic groups of small dimension. Some Dynkin diagrams of

small ranks coincide:

(26.20) A1 = B 1 = C 1

(26.21) D2 = A 1 + A 1

(26.22) B2 = C 2

(26.23) A3 = D 3

We describe explicitly the corresponding isomorphisms for adjoint groups (ana-

logues for algebras are in §??):

A1 = B1 = C1 . Let (V, q) be a regular quadratic form of dimension 3 over a

¬eld F . Then C0 (V, q) is a quaternion algebra over F . The canonical homomor-

phism

O+ (V, q) ’ PGL1 C0 (V, q) = PGSp C0 (V, q), σ q

is injective (see §??) and hence is an isomorphism of adjoint simple groups of

types B1 , A1 and C1 since by dimension count its image has the same dimension

374 VI. ALGEBRAIC GROUPS

as the target group, and since these groups are connected they must coincide, by

Propositions (??) and (??). (We will use this argument several times below.)

Let Q be a quaternion algebra over F and let Q0 = { x ∈ Q | TrdQ (x) = 0 }.

For x ∈ Q0 , we have x2 ∈ F , and the squaring map s : Q0 ’ F is a quadratic form

of discriminant 1 on Q0 (see §??). Consider the conjugation homomorphism

f : GL1 (Q) ’ O+ (Q0 , s).

Since Q0 generates Q, ker(f ) = Gm and the injection

PGSp(Q, σ) = PGL1 (Q) ’ O+ (Q0 , s)

is an isomorphism of adjoint simple groups of types C1 , A1 , and B1 .

D2 = A1 + A1 . Let A be a central simple algebra over F of degree 4 with a

quadratic pair (σ, f ). Then C(A, σ, f ) is a quaternion algebra over a quadratic ´tale

e

extension Z of F . We have the canonical injection

PGO+ (A, σ, f ) ’ AutZ C(A, σ, f ) = RZ/F PGL1 C(A, σ, f )

which is an isomorphism between adjoint groups of type D2 and those of type

A1 + A 1 .

Conversely, let Q be a quaternion algebra over an ´tale quadratic extension

e

Z/F . The norm A = NZ/F (Q) is a central simple algebra of degree 4 over F with

a canonical quadratic pair (σ, f ) (see §??). We have the natural homomorphism

g : RZ/F GL1 (Q) ’ GO+ (A, σ, f ), x ∈ Q — ’ x — x ∈ A— .

R R

—

One checks that x — x ∈ R— if and only if x ∈ ZR , hence g ’1 (Gm ) = RZ/F (Gm,Z ).

By factoring out these subgroups we obtain an injective homomorphism

RZ/F PGL1 (Q) ’ PGO+ (A, σ, f )

which is actually an isomorphism from an adjoint group of type A1 + A1 to one of

type D2 .

B2 = C2 . Let (V, q) be a regular quadratic form of dimension 5. Then C0 (V, q)

is a central simple algebra of degree 4 with (canonical) symplectic involution „ .

There is a canonical injective homomorphism (see §??)

O+ (V, q) ’ PGSp C0 (V, q), „

which is in fact an isomorphism from an adjoint simple groups of type B2 to one

of type C2 .

Conversely, for a central simple algebra A of degree 4 over F with a symplectic

involution σ, the F -vector space

Symd(A, σ)0 = { x ∈ Symd(A, σ) | TrpA (x) = 0 }

admits the quadratic form sσ (x) = x2 ∈ F (see §??). Consider the conjugation

homomorphism

f : GSp(A, σ) ’ O+ Symd(A, σ)0 , sσ , a ’ Int(a).

Since Symd(A, σ) generates A, one has ker(f ) = Gm . Hence, the injection

PGSp(A, σ) ’ O+ V, q

is an isomorphism from an adjoint simple group of type C2 to one of type B2 .

§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 375

A3 = D3 . Let A be a central simple algebra of degree 6 over F with an orthog-

onal pair (σ, f ). Then C(A, σ, f ) is a central simple algebra of degree 4 over an ´tale

e

quadratic extension Z/F with a unitary involution σ. The natural homomorphism

PGO+ (A, σ, f ) ’ PGU C(A, σ, f ), σ

is injective (see §??) and hence is an isomorphism from an adjoint simple group of

type D3 to one of type A3 .

Conversely, let B be a central simple algebra of degree 4 over an ´tale quadratic

e

extension Z/F with a unitary involution „ . Then the discriminant algebra D(B, „ )

is a central simple algebra of degree 6 over F with canonical quadratic pair („ , f ).

Consider the natural homomorphism

GU(B, „ ) ’ GO+ D(B, „ ), „ , f .

One checks (in the split case) that g ’1 (Gm ) = GL1 (Z). By factoring out these

subgroups we obtain an injection

PGU(B, „ ) ’ PGO+ D(B, „ ), „ , f

which is an isomorphism from an adjoint simple group of type A3 to one of type D3 .

§27. Tits Algebras of Semisimple Groups

The Cli¬ord algebra, the discriminant algebra, the »-powers of a central simple

algebra all arise as to be so-called Tits algebras of the appropriate semisimple

groups. In this section we de¬ne Tits algebras and classify them for simple groups

of the classical series.

For this we need some results on the classi¬cation of representations of split

semisimple groups. Let G be a split semisimple group over F . Choose a split

maximal torus T ‚ G. Fix a system of simple roots in ¦(G), so we have the

corresponding cone Λ+ ‚ Λ of dominant weights.

Let ρ : G ’ GL(V ) be a representation. By the representation theory of diag-

onalizable groups (??) one can associate to the representation ρ|T a ¬nite number

of weights, elements of T — . If ρ is irreducible, among the weights there is a largest

(with respect to the ordering on Λ). It lies in Λ+ and is called the highest weight

of ρ (Humphreys [?]).

(27.1) Theorem. The map

Isomorphism classes of

T — © Λ+

←’

irreducible representations of G

taking the class of a representation ρ to its highest weight, is a bijection.

Reference: Tits [?, Th.2.5]

(27.2) Remark. If G is a simply connected group (i.e., T — = Λ), then T — © Λ+ =

Λ+ .

(27.3) Remark. The classi¬cation of irreducible representations of a split semi-

simple groups does not depend on the base ¬eld in the sense that an irreducible

representation remains irreducible over an arbitrary ¬eld extension and any irre-

ducible representation over an extension comes from the base ¬eld.

376 VI. ALGEBRAIC GROUPS

27.A. De¬nition of the Tits algebras. Let G be a semisimple (not neces-

sarily split) group over F and let T ‚ G be a maximal torus. Choose a system of

—

simple roots Π ‚ ¦ = ¦(G). The group “ acts on Tsep and is the identity on ¦, Λ,

Λr (but not Π).

—

There is another action of “ on Tsep , called the —-action, which is de¬ned as

follows. Take any γ ∈ “. Since the Weyl group W acts simply transitively on the

set of systems of simple roots and γΠ is clearly a system of simple roots, there is a

unique w ∈ W such that w(γΠ) = Π. We set γ — ± = w(γ±) ∈ Π for any ± ∈ Π.

This action, de¬ned on Π, extends to an action on Λ which is the identity on Π, ¦,

Λr , Λ+ . Note that since W acts trivially on Λ/Λr , the —-action on Λ/Λr coincides

with the usual one.

Choose a ¬nite Galois extension F ‚ L ‚ Fsep splitting T and hence G. The —-

action of “ then factors through Gal(L/F ). Let ρ : GL ’ GL(V ) be an irreducible

representation over L (so V is an L-vector space) with highest weight » ∈ Λ+ © T —

(see Theorem (??)). For any γ ∈ “ we can de¬ne the L-space γ V as V as an abelian

group and with the L-action x —¦ v = γ ’1 (x) · v, for all x ∈ L, v ∈ V . Then v ’ v

viewed as a map V ’ γ V is γ-semilinear. Denote it iγ .

Let A = F [G] and let ρ : V ’ V —L AL be the comodule structure for ρ (see

p. ??). The composite

i’1 iγ —(γ—Id)

ρ

γ

γV ’ ’ V ’ V —L (L —F A) ’ ’ ’ ’ γ V —L (L —F A)

’ ’ ’ ’ ’’

gives the comodule structure for some irreducible representation

γρ: GL ’ GL(γ V ).

(Observe that the third map is well-de¬ned because both iγ and γ — Id are γ-

semilinear.) Clearly, the weights of γ ρ are obtained from the weights of ρ by ap-

plying γ. Hence, the highest weight of γ ρ is γ — ».

Assume now that » ∈ Λ+ © T — is invariant under the —-action. Consider the

conjugation representation

g ’ ± ’ ρ(g) —¦ ± —¦ ρ(g)’1 .

Int(ρ) : G ’ GL EndF (V ) ,

Let EndG (V ) be the subalgebra of G-invariant elements in EndF (V ) under Int(ρ).

Then

(27.4) EndG (V ) —F L EndGL (V —F L) EndGL γ∈Gal(L/F ) γ V

since V —F L is L-isomorphic to γ V via v — x ’ (γ ’1 x · v)γ . Since the represen-

tation γ ρ is of highest weight γ — » = », it follows from Theorem (??) that γ ρ ρ,

i.e., all the G-modules γ V are isomorphic to V . Hence, the algebras in (??) are

isomorphic to

EndGL (V n ) = Mn EndGL (V )

where n = [L : F ].

(27.5) Lemma. EndGL (V ) L.

Proof : Since ρ is irreducible, EndGL (V ) is a division algebra over L by Schur™s

lemma. But ρalg remains irreducible by Remark (??), hence EndGL (V ) —L Falg is

also a division algebra and therefore EndGL (V ) = L.

§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 377

It follows from the lemma that EndG (V ) —F L Mn (L), hence EndG (V ) is a

central simple algebra over F of degree n. Denote its centralizer in EndF (V ) by

A» . This is a central simple algebra over F of degree dimL V . It is clear that A»

is independent of the choice of L. The algebra A» is called the Tits algebra of the

group G corresponding to the dominant weight ».

Since the image of ρ commutes with EndG (V ), it actually lies in A» . Thus we

obtain a representation

ρ : G ’ GL1 (A» ).

By the double centralizer theorem (see (??)), the centralizer of EndG (V ) —F L

in EndF (V ) —F L is A» —F L. On the other hand it contains EndL (V ) (where the

image of ρ lies). By dimension count we have

A» —F L = EndL (V )

and hence the representation (ρ )L is isomorphic to ρ. Thus, ρ can be considered

as a descent of ρ from L to F . The restriction of ρ to the center C = C(G) ‚ G

is given by the restriction of » on C, i.e., is the character of the center C given by

the class of » in C — = T — /Λr ‚ Λ/Λr .

The following lemma shows the uniqueness of the descent ρ .

(27.6) Lemma. Let µi : G ’ GL1 (Ai ), i = 1, 2 be two homomorphisms where

the Ai are central simple algebras over F . Assume that the representations (µ i )sep

are isomorphic and irreducible. Then there is an F -algebra isomorphism ± : A 1 ’

A2 such that GL1 (±) —¦ µ1 = µ2 .

Proof : Choose a ¬nite Galois ¬eld extension L/F splitting G and the Ai , Ai —F L

∼

EndL (Vi ). An L-isomorphism V1 ’ V2 of GL -representations gives rise to an

’

algebra isomorphism

∼

± : EndF (V1 ) ’ EndF (V2 )

’

taking EndG (V1 ) to EndG (V2 ). Clearly, Ai lies in the centralizer of EndG (Vi ) in

EndF (Vi ). By dimension count Ai coincides with the centralizer, hence ±(A1 ) =

A2 .

Let π : G ’ G be a central isogeny with G simply connected. Then the Tits

algebra built out of a representation ρ of GL is the Tits algebra of the group GL

corresponding to the representation ρ —¦ πL . Hence, in order to classify Tits algebras

one can restrict to simply connected groups.

Assume that G is a simply connected semisimple group. For any » ∈ Λ/Λr

consider the corresponding (unique) minimal weight χ(») ∈ Λ+ . The uniqueness

shows that χ(γ») = γ — χ(») for any γ ∈ “. Hence, if » ∈ (Λ/Λr )“ , then clearly

χ(») ∈ Λ“ (with respect to the —-action); the Tits algebra Aχ(») is called a minimal

+

Tits algebra and is denoted simply by A» . For example, if » = 0, then A» = F .

(27.7) Theorem. The map

β : (Λ/Λr )“ ’ Br F, » ’ [A» ]

is a homomorphism.

Reference: Tits [?, Cor. 3.5].

378 VI. ALGEBRAIC GROUPS

If » ∈ Λ/Λr is not necessarily “-invariant, let

“0 = { γ ∈ “ | γ(») = » } ‚ “

and F» = (Fsep )“0 . Then » ∈ (Λ/Λr )“0 and one gets a Tits algebra A» , which is a

central simple algebra over F» , for the group GF» . The ¬eld F» is called the ¬eld

of de¬nition of ».

27.B. Simply connected classical groups. We give here the classi¬cation

of the minimal Tits algebras of the absolutely simple simply connected groups of

classical type.

Type An , n ≥ 1. Let ¬rst G = SL1 (A) where A is a central simple algebra of

degree n + 1 over F . Then C = µn+1 , C — = Z/(n + 1)Z with the trivial “-action.

For any i = 0, 1, . . . , n, consider the natural representation

ρi : G ’ GL1 (»i A).

In the split case ρi is the i-th exterior power representation with the highest weight

e1 + e2 + · · · + ei in the notation of §??, which is a minimal weight. Hence, the

»-powers »i A, for i = 0, 1, . . . , n, (see §??) are the minimal Tits algebras of G.

Now let G = SU(B, „ ) where B is a central simple algebra of degree n + 1 with

a unitary involution over a quadratic separable ¬eld extension K/F . The group

“ acts on C — = Z/(n + 1)Z by x ’ ’x through Gal(K/F ). The only nontrivial

element in (C — )“ is » = n+1 + (n + 1)Z (n should be odd). There is a natural

2

homomorphism

ρ : G ’ GL1 D(B, „ )

which in the split case is the external n+1 -power. Hence, the discriminant algebra

2

(see §??) D(B, „ ) is the minimal Tits algebra corresponding to » for the group G.

The ¬elds of de¬nition Fµ of the other nontrivial characters µ = i + (n + 1)Z ∈

C — , (i = (n+1) ), coincide with K. Hence, by extending the base ¬eld to K one sees

2

that Aµ »i B.

Type Bn , n ≥ 1. Let G = Spin(V, q), here (V, q) is a regular quadratic form

of dimension 2n + 1. Then C = µ2 , C — = Z/2Z = {0, »}. The embedding

G ’ GL1 C0 (V, q)

in the split case is the spinor representation with highest weight 1 (e1 +e2 +· · ·+en )

2

in the notation of §??, which is a minimal weight. Hence, the even Cli¬ord algebra

C0 (V, q) is the minimal Tits algebra A» .

Type Cn , n ≥ 1. Let G = Sp(A, σ) where A is a central simple algebra of

degree 2n with a symplectic involution σ. Then C = µ2 , C — = Z/2Z = {0, »}. The

embedding

G ’ GL1 (A)

in the split case is the representation with highest weight e1 in the notation of §??,

which is a minimal weight. Hence, A is the minimal Tits algebra A» .

§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 379

Type Dn , n ≥ 2, n = 4. Let G = Spin(A, σ, f ) where A is a central simple

algebra of degree 2n with a quadratic pair (σ, f ), C — = {0, », »+ , »’ } where »

factors through O+ (A, σ, f ). The composition

Spin(A, σ, f ) ’ GO+ (A, σ, f ) ’ GL1 (A)

in the split case is the representation with highest weight e1 in the notation of §??,

which is a minimal weight. Hence, A is the minimal Tits algebra A» .

Assume further that the discriminant of σ is trivial (i.e., the center Z of the

Cli¬ord algebra is split). The group “ then acts trivially on C — . The natural

compositions

Spin(A, σ, f ) ’ GL1 C(A, σ, f ) ’ GL1 C ± (A, σ, f )

in the split case are the representations with highest weights 1 (e1 + · · · + en’1 ± en )

2

which are minimal weights. Hence, C ± (A, σ, f ) are the minimal Tits algebras A»± .

If disc(σ) is not trivial then “ interchanges »+ and »’ , hence the ¬eld of

de¬nition of »± is Z. By extending the base ¬eld to Z one sees that A»± =

C(A, σ, f ). Again, the case of D4 is exceptional, because of triality, and we give on

p. ?? a description of the minimal Tits algebra in this case.

27.C. Quasisplit groups. A semisimple group G is called quasisplit if there

is a maximal torus T ‚ G and a system Π of simple roots in the root system ¦

of G with respect to T which is “-invariant with respect to the natural action, or

—

equivalently, if the —-action on Tsep coincides with the natural one. For example,

split groups are quasisplit.

Let G be a quasisplit semisimple group. The natural action of “ on the

system Π of simple roots, which is invariant under “, de¬nes an action of “ on

Dyn(G) = Dyn(¦) by automorphisms of the Dynkin diagram. Simply connected

and adjoint split groups are classi¬ed by their Dynkin diagrams. The following

statement generalizes this result for quasisplit groups.

(27.8) Proposition. Two quasisplit simply connected (resp. adjoint) semisimple

groups G and G are isomorphic if and only if there is a “-bijection between Dyn(G)

and Dyn(G ). For any Dynkin diagram D and any (continuous) “-action on D

there is a quasisplit simply connected (resp. adjoint) semisimple group G and a

“-bijection between Dyn(G) and D.

—

The “-action on Dyn(G) is trivial if and only if “ acts trivially on Tsep , hence T

and G are split. Therefore, if Aut(Dyn(G)) = 1 (i.e., Dyn(G) has only irreducible

components Bn , Cn , E7 , E8 , F4 , G2 ) and G is quasisplit, then G is actually split.

(27.9) Example. The case An , n > 1. A non-trivial action of the Galois group

“ on the cyclic group Aut(An ) of order two factors through the Galois group of

a quadratic ¬eld extension L/F . The corresponding quasisplit simply connected

simple group of type An is isomorphic to SU(V, h), where (V, h) is a non-degenerate

hermitian form over L/F of dimension n + 1 and maximal Witt index.

(27.10) Example. The case Dn , n > 1, n = 4. As in the previous example, to give

a nontrivial “-action on Dn is to give a quadratic Galois ¬eld extension L/F . The

corresponding quasisplit simply connected simple group of type Dn is isomorphic

to Spin(V, q), where (V, q) is a non-degenerate quadratic form of dimension 2n and

Witt index n ’ 1 with the discriminant quadratic extension L/F .

380 VI. ALGEBRAIC GROUPS

Exercises

1. If L is an ´tale F -algebra, then Autalg (L) is an ´tale group scheme correspond-

e e

ing to the ¬nite group AutFsep (Lsep ) with the natural Gal(Fsep /F )-action.

2. Let G be an algebraic group scheme. Prove that the following statements are

equivalent:

(a) G is ´tale,

e

0

(b) G = 1,

(c) G is smooth and ¬nite,

(d) Lie(G) = 0.

3. Prove that Hdiag is algebraic if and only if H is a ¬nitely generated abelian

group.

4. Let H be a ¬nitely generated abelian group, and let H ‚ H be the subgroup

of elements of order prime to char F . Prove that (Hdiag )0 (H/H )diag and

π0 (Hdiag ) Hdiag .