Orthogonal groups. Let (V, q) be a quadratic form of dimension n over F

and let bq be the polar bilinear form of q on V . We recall that the form q is regular

if bq is a nonsingular bilinear form except for the case n is odd and char F = 2.

In this case bq is symplectic and is degenerate. The radical of q is the space V ⊥

and (in case charF = 2 and dimF V is odd) q is regular if dim rad(bq ) = 1, say

rad(q) = F · v, q(v) = 0.

We view q as an element of S 2 (V — ), the space of degree 2 elements in the

symmetric algebra S 2 (V — ). There is a natural representation

ρ : GL(V ) ’ GL S 2 (V — ) .

We set O(V, q) for the group AutGL(V ) (q) and call it the orthogonal group of (V, q):

O(V, q)(R) = { ± ∈ GL(VR ) | qR (±v) = qR (v) for v ∈ VR }.

The associated classical algebraic group has dimension n(n’1) (Borel [?, 23.6]). For

2

± ∈ End(V ), we have 1 + ± · µ ∈ O(V, q) if and only if bq (v, ±v) = 0 for all v ∈ V .

Hence

Lie O(V, q) = { ± ∈ End(V ) | bq (v, ±v) = 0 for v ∈ V } = o(V, q).

The dimensions are:

n(n’1)

if n is even or char F = 2,

2

dim Lie O(V, q) = n(n’1)

+ 1 if n is odd and char F = 2.

2

Hence, in the ¬rst case O(V, q) is a smooth group scheme. We consider now the

following cases:

(a) char F = 2 and n is even: we de¬ne

∆

O+ (V, q) = ker O(V, q) ’ Z/2Z

’

where ∆ is the Dickson invariant, i.e., ∆(±) = 0 for ± ∈ O(V, q)(R) if

± induces the identity automorphism of the center of the Cli¬ord algebra

and ∆(±) = 1 if not (see (??)). The associated classical algebraic group

is known to be connected (Borel [?, 23.6]). Hence, O+ (V, q) is a smooth

connected group scheme.

(b) char F = 2 or n is odd: we set

det

O+ (V, q) = ker O(V, q) ’’ Gm

’

where det is the determinant map. Here also the associated classical alge-

braic group is known to be connected (Borel [?]).

We get in each case

Lie O+ (V, q) = { ± ∈ End(V ) | tr(±) = 0, bq (v, ±v) = 0 for v ∈ V }.

If char F = 2 this Lie algebra coincides with Lie O(V, q) and O+ (V, q) is the

connected component of O(V, q). If char F = 2, then Lie O+ (V, q) Lie O(V, q)

hence

n(n ’ 1)

dim Lie O+ (V, q) =

2

and O (V, q) is a smooth connected group scheme. Thus in every case O+ (V, q) is

+

a connected algebraic group. Consider the conjugation homomorphism

GL1 C0 (V, q) ’ GL C(V, q) , x ’ Int(x)

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 349

where C(V, q) = C0 (V, q) • C1 (V, q) is the Cli¬ord algebra. The inverse image of

the normalizer NV of the subspace V ‚ C(V, q) is “+ (V, q), the even Cli¬ord group

of (V, q),

“+ (V, q)(R) = { g ∈ C0 (V, q)— | q · VR · g ’1 = VR }.

R

It follows from Example (??.??) that

Lie “+ (V, q) = { x ∈ C0 (V, q) | [x, V ] ‚ V } = V · V ‚ C0 (V, q).

Let

χ : “+ (V, q) ’ O+ (V, q), x ’ Int(x)|V .

Clearly, ker χ = Gm ‚ “+ (V, q). Since χalg is surjective, we have by Proposi-

tion (??) an exact sequence

χ

1 ’ Gm ’ “+ (V, q) ’ O+ (V, q) ’ 1.

’

Hence by Corollary (??) “+ (V, q) is smooth.

The kernel of the spinor norm homomorphism

Sn : “+ (V, q) ’ Gm , x ’ x · σ(x)

is the spinor group of (V, q) and is denoted Spin(V, q). Thus,

Spin(V, q)(R) = { g ∈ C0 (V, q)— | g · VR · g ’1 = VR , g · σ(g) = 1 }

R

The di¬erential d(Sn) is given by

d(Sn)(uv) = uv + σ(uv) = uv + vu = bq (u, v).

In particular, Sn is separable and

Lie Spin(V, q) = { x ∈ V · V ‚ C0 (V, q) | x + σ(x) = 0 }.

Since Snalg is surjective, we have by Proposition (??) an exact sequence

Sn

1 ’ Spin(V, q) ’ “+ (V, q) ’ Gm ’ 1.

’

Hence by Proposition (??) Spin(V, q) is smooth. The classical algebraic group asso-

ciated to Spin(V, q) is known to be connected (Borel [?, 23.3]), therefore Spin(V, q)

is connected.

The kernel of the composition

χ

f : Spin(V, q) ’ “+ (V, q) ’ O+ (V, q)

’

is µ2 . Since falg is surjective, it follows by Proposition (??) that f is surjective.

Hence, f is a central isogeny and

O+ (V, q) Spin(V, q)/µ2 .

(23.5) Remark. The preceding discussion focuses on orthogonal groups of quad-

ratic spaces. Orthogonal groups of symmetric bilinear spaces may be de¬ned in a

similar fashion: every nonsingular symmetric nonalternating bilinear form b on a

vector space V may be viewed as an element of S 2 (V )— , and letting GL(V ) act on

S 2 (V )— we may set O(V, b) = AutGL(V ) (b).

If char F = 2 we may identify S 2 (V )— to S 2 (V — ) by mapping every symmetric

bilinear form b to its associated quadratic form qb de¬ned by qb (x) = b(x, x), hence

O(V, b) = O(V, qb ). If char F = 2 the group O(V, b) is not smooth, and if F is not

perfect there may be no associated smooth algebraic group, see Exercise ??.

350 VI. ALGEBRAIC GROUPS

Suppose F is perfect of characteristic 2. In that case, there is an associated

smooth algebraic group O(V, b)red . If dim V is odd, O(V, b)red turns out to be

isomorphic to the symplectic group of an alternating space of dimension dim V ’ 1,

see Exercise ??. If dim V is even, O(V, b)red contains a nontrivial solvable connected

normal subgroup, see Exercise ??; it is therefore not semisimple (see §?? for the

de¬nition of semisimple group).

23.B. Quadratic pairs. Let A be a central simple algebra of degree n = 2m

over F , and let (σ, f ) be a quadratic pair on A. Consider the homomorphism

Aut(A, σ) ’ GL Sym(A, σ)— , ± ’ (g ’ g —¦ ±).

The inverse image of the stabilizer Sf of f is denoted PGO(A, σ, f ) and is called

the projective orthogonal group:

PGO(A, σ, f )(R) = { ± ∈ Aut(A, σ) | fR —¦ ± = fR }.

If R satis¬es the SN -condition, then, setting (A, σ)+ = Sym(A, σ),

PGO(A, σ, f )(R) =

{ a ∈ A— | a · σR (a) ∈ R— , f (axa’1 ) = f (x) for x ∈ (AR , σR )+ }/R— .

R

In the split case A = End(V ), with q a quadratic form corresponding to the

quadratic pair (σ, f ), we write PGO(V, q) for this group. The inverse image of

PGO(A, σ, f ) under

Int : GL1 (A) ’ Autalg (A)

is the group of orthogonal similitudes and is denoted GO(A, σ, f ):

GO(A, σ, f )(R) =

{ a ∈ A— | a · σR (a) ∈ R— , f (axa’1 ) = f (x) for x ∈ (AR , σR )+ }.

R

One sees that 1 + a · µ ∈ GO(A, σ, f )(F [µ]) if and only if a + σ(a) ∈ F and

f (ax ’ xa) = 0 for all symmetric x. Thus

Lie GO(A, σ, f ) = { a ∈ A | a + σ(a) ∈ F , f (ax ’ xa) = 0 for x ∈ Sym(A, σ) }.

An analogous computation shows that

Lie PGO(A, σ, f ) = Lie GO(A, σ, f ) /F.

The kernel of the homomorphism

µ : GO(A, σ, f ) ’ Gm , a ’ a · σ(a)

is denoted O(A, σ, f ) and is called the orthogonal group,

O(A, σ, f )(R) = { a ∈ A— | a · σ(a) = 1, f (axa’1 ) = f (x) for x ∈ Sym(A, σ)R }.

R

Since for a ∈ A with a + σ(a) = 0 one has f (ax ’ xa) = f ax + σ(ax) = Trd(ax),

it follows that the condition f (ax ’ xa) = 0 for all x ∈ Sym(A, σ) is equivalent

to a ∈ Alt(A, σ). Thus

Lie O(A, σ, f ) = Alt(A, σ)

(and does not depend on f !).

In the split case we have O(A, σ, f ) = O(V, q), hence by ??, O(A, σ, f ) is

smooth.

The sequence

µ

1 ’ O(A, σ, f ) ’ GO(A, σ, f ) ’ Gm ’ 1

’

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 351

is exact by Proposition (??), since µalg is surjective. It follows from Corollary (??)

that GO(A, σ, f ) is smooth. By Proposition (??), the natural homomorphism

GO(A, σ, f ) ’ PGO(A, σ, f ) is surjective, hence PGO(A, σ, f ) is smooth. There

is an exact sequence

1 ’ Gm ’ GO(A, σ, f ) ’ PGO(A, σ, f ) ’ 1.

The kernel of the composition

g : O(A, σ, f ) ’ GO(A, σ, f ) ’ PGO(A, σ, f )

is µ2 . Clearly, galg is surjective, hence g is surjective. Therefore, g is a central

isogeny and

PGO(A, σ, f ) O(A, σ, f )/µ2 .

Now comes into play the Cli¬ord algebra C(A, σ, f ). By composing the natural

homomorphism

PGO(A, σ, f ) ’ Autalg C(A, σ, f )

with the restriction map

Autalg C(A, σ, f ) ’ Autalg (Z) = Z/2Z

where Z is the center of C(A, σ, f ), we obtain a homomorphism PGO(A, σ, f ) ’

Z/2Z, the kernel of which we denote PGO+ (A, σ, f ). The inverse image of this

group in GO(A, σ, f ) is denoted GO+ (A, σ, f ) and the intersection of GO+ (A, σ, f )

with O(A, σ, f ) by O+ (A, σ, f ). In the split case O+ (A, σ, f ) = O+ (V, q), hence

O+ (A, σ, f ) is smooth and connected. In particular it is the connected component

of O(A, σ, f ). It follows from the exactness of

1 ’ µ2 ’ O+ (A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

that PGO+ (A, σ, f ) is also a connected algebraic group, namely the connected

component of PGO(A, σ, f ).

Let B(A, σ, f ) be the Cli¬ord bimodule. Consider the representation

c ’ x ’ (c — x · c’1 ) .

GL1 C(A, σ, f ) ’ GL B(A, σ, f ) ,

Let b : A ’ B(A, σ, f ) be the canonical map. Let “(A, σ, f ) be the inverse image

of the normalizer Nb(A) of the subspace b(A) ‚ B(A, σ, f ) and call it the Cli¬ord

group of (A, σ, f ),

“(A, σ, f )(R) = { c ∈ C(A, σ, f )— | c — b(A)R · c’1 = b(A)R }.

R

In the split case “(A, σ, f ) = “+ (V, q) is a smooth group and

Lie “(A, σ, f ) = V · V = c(A) ‚ C(A, σ, f ).

Hence “(A, σ, f ) is a smooth algebraic group and

Lie “(A, σ, f ) = c(A).

For any g ∈ “(A, σ, f )(R) one has g · σ(g) ∈ R— , hence there is a spinor norm

homomorphism

Sn : “(A, σ, f ) ’ Gm , g ’ g · σ(g).

We denote the kernel of Sn by Spin(A, σ, f ) and call it the spinor group of (A, σ, f ).

It follows from the split case (where Spin(A, σ, f ) = Spin(V, q)) that Spin(A, σ, f )

is a connected algebraic group.

352 VI. ALGEBRAIC GROUPS

Let χ : “(A, σ, f ) ’ O+ (A, σ, f ) be the homomorphism de¬ned by the formula

c’1 — (1)b · c = χ(c) · b, and let g be the composition

χ

Spin(A, σ, f ) ’ “(A, σ, f ) ’ O+ (A, σ, f ).

’

Clearly, ker g = µ2 and, since galg is surjective, g is surjective, hence g is a central

isogeny and

O+ (A, σ, f ) Spin(A, σ, f )/µ2 .

Consider the natural homomorphism

C : PGO+ (A, σ, f ) ’ AutZ C(A, σ, f ), σ .

If n = deg A with n > 2, then c(A)R generates the R-algebra C(A, σ, f )R for any

R ∈ Alg F . Hence CR is injective and C is a closed embedding by Proposition (??).

By (??), there is an exact sequence

Int

1 ’ GL1 (Z) ’ Sim C(A, σ, f ), σ ’’ AutZ C(A, σ, f ), σ ’ 1.

’

Let „¦(A, σ, f ) be the group Int’1 (im C), which we call the extended Cli¬ord group.

Note that “(A, σ, f ) ‚ „¦(A, σ, f ) ‚ Sim C(A, σ, f ), σ . By Proposition (??) we

have a commutative diagram with exact rows:

O+ (A, σ, f )

1 ’’’

’’ Gm ’ ’ ’ “(A, σ, f ) ’ ’ ’

’’ ’’ ’’’ 1

’’

¦ ¦ ¦

¦ ¦ ¦

1 ’ ’ ’ GL1 (Z) ’ ’ ’ „¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f ) ’ ’ ’ 1.

’’ ’’ ’’ ’’

The ¬rst two vertical maps are injective. By Corollary (??), the group „¦(A, σ, f )

is smooth.

(23.6) Remark. If char F = 2, the involution σ is orthogonal and f is prescribed.

We then have,

O(A, σ, f ) = O(A, σ)

GO(A, σ, f ) = GO(A, σ)

PGO(A, σ, f ) = PGO(A, σ).

§24. Root Systems

In this section we recall basic results from the theory of root systems and refer

to Bourbaki [?] for details. Let V be an R-vector space of positive ¬nite dimension.

An endomorphism s ∈ End(V ) is called a re¬‚ection with respect to ± ∈ V , ± = 0 if

(a) s(±) = ’±,

(b) there is a hyperplane W ‚ V such that s|W = Id.

We will use the natural pairing

V — — V ’ R, χ — v ’ χ, v = χ(v).

A re¬‚ection s with respect to ± is given by the formula s(v) = v ’ χ, v ± for a

uniquely determined linear form χ ∈ V — with χ|W = 0 and χ, ± = 2. Note that

a ¬nite set of vectors which spans V is preserved as a set by at most one re¬‚ection

with respect to any given ± (see Bourbaki [?, Chapter VI, § 1, Lemme 1]).

A ¬nite subset ¦ ‚ V = 0 is called a (reduced ) root system if

(a) 0 ∈ ¦ and ¦ spans V .

§24. ROOT SYSTEMS 353

(b) If ± ∈ ¦ and x± ∈ ¦ for x ∈ R, then x = ±1.

(c) For each ± ∈ ¦ there is a re¬‚ection s± with respect to ± such that s± (¦) =

¦.

(d) For each ±, β ∈ ¦, s± (β) ’ β is an integral multiple of ±.

The elements of ¦ are called roots. The re¬‚ection s± in (??) is uniquely determined

by ±. For ± ∈ ¦, we de¬ne ±— ∈ V — by

s± (v) = v ’ ±— , v · ±.

Such ±— are called coroots. The set ¦— = {±— ∈ V } forms the dual root system

in V — . Clearly, ±— , β ∈ Z for any ±, β ∈ ¦ and ±— , ± = 2.

An isomorphism of root systems (V, ¦) and (V , ¦ ) is an isomorphism of vector

spaces f : V ’ V such that f (¦) = ¦ . The automorphism group Aut(V, ¦) is a

¬nite group. The subgroup W (¦) of Aut(V, ¦) generated by all the re¬‚ections s± ,

± ∈ ¦, is called the Weyl group of ¦.

Let ¦i be a root system in Vi , i = 1, 2, . . . , n, and V = V1 • V2 • · · · • Vn ,

¦ = ¦1 ∪ ¦2 ∪ · · · ∪ ¦n . Then ¦ is a root system in V , called the sum of the ¦i .

We write ¦ = ¦1 + ¦2 + · · · + ¦n . A root system ¦ is called irreducible if ¦ is not

isomorphic to the sum ¦1 + ¦2 of some root systems. Any root system decomposes

uniquely into a sum of irreducible root systems.

Let ¦ be a root system in V . Denote by Λr the (additive) subgroup of V

generated by all roots ± ∈ ¦; Λr is a lattice in V , called the root lattice. The lattice

Λ = { v ∈ V | ±— , v ∈ Z for ± ∈ ¦ }

in V , dual to the root lattice generated by ¦— ‚ V — , is called the weight lattice.

Clearly, Λr ‚ Λ and Λ/Λr is a ¬nite group. The group Aut(V, ¦) acts naturally on

Λ, Λr , and Λ/Λr , and W (¦) acts trivially on Λ/Λr .

A subset Π ‚ ¦ of the root system ¦ is a system of simple roots or a base of a

root system if for any ± ∈ ¦,

±= nβ · β

β∈Π

for some uniquely determined nβ ∈ Z and either nβ ≥ 0 for all β ∈ Π or nβ ¤ 0 for

all β ∈ Π. In particular, Π is a basis of V . For a system of simple roots Π ‚ ¦ and

w ∈ W (¦) the subset w(Π) is also a system of simple roots. Every root system has

a base and the Weyl group W (¦) acts simply transitively on the set of bases of ¦.

Let ¦ be a root system in V and Π ‚ ¦ be a base. We de¬ne a graph, called

the Dynkin diagram of ¦, which has Π as its set of vertices. The vertices ± and

β are connected by ±— , β · β — , ± edges. If ±— , β > β — , ± , then all the edges

between ± and β are directed, with ± the origin and β the target. This graph does

not depend (up to isomorphism) on the choice of a base Π ‚ ¦, and is denoted

Dyn(¦). The group of automorphisms of Dyn(¦) embeds into Aut(V, ¦), and

Aut(V, ¦) is a semidirect product of W (¦) (a normal subgroup) and Aut Dyn(¦) .

In particular, Aut Dyn(¦) acts naturally on Λ/Λr .

Two root systems are isomorphic if and only if their Dynkin diagrams are iso-

morphic. A root system is irreducible if and only if its Dynkin diagram is connected.

The Dynkin diagram of a sum ¦1 + · · · + ¦n is the disjoint union of the Dyn(¦i ).

Let Π ‚ ¦ be a system of simple roots. The set

Λ+ = { χ ∈ Λ | ±— , χ ≥ 0 for ± ∈ Π }

354 VI. ALGEBRAIC GROUPS

is the cone of dominant weights in Λ (relative to Π). We introduce a partial ordering

on Λ: χ > χ if χ ’ χ is sum of simple roots. For any » ∈ Λ/Λr there exists a

unique minimal dominant weight χ(») ∈ Λ+ in the coset ». Clearly, χ(0) = 0.

24.A. Classi¬cation of irreducible root systems. There are four in¬nite

families An , Bn , Cn , Dn and ¬ve exceptional irreducible root systems E6 , E7 , E8 ,

F4 , G2 . We refer to Bourbaki [?] for the following datas about root systems.

Type An , n ≥ 1. Let V = Rn+1 /(e1 + e2 + · · · + en+1 )R where {e1 , . . . , en+1 }

is the canonical basis of Rn+1 . We denote by ei the class of ei in V .

Root system : ¦ = { ei ’ ej | i = j }, n(n + 1) roots.

Root lattice : Λr = { ai ei | ai = 0 }.

ei Z, Z/(n + 1)Z.

Weight lattice : Λ= Λ/Λr

Simple roots : Π = {e1 ’ e2 , e2 ’ e3 , . . . , en ’ en+1 }.

Dynkin diagram : c c ppp c

1 2 n

Aut Dyn(¦) : {1} if n = 1, {1, „ } if n ≥ 2.

Dominant weights : Λ+ = { ai · ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an+1 }.

Minimal weights : e1 + e2 + · · · + ei , i = 1, 2, . . . , n + 1.

Type Bn , n ≥ 1. Let V = Rn with canonical basis {ei }.

: ¦ = { ±ei , ±ei ± ej | i > j }, 2n2 roots.

Root system

: Λ r = Zn .

Root lattice

: Λ = Λr + 1 (e1 + e2 + · · · + en )Z, Z/2Z.

Weight lattice Λ/Λr

2

Simple roots : Π = {e1 ’ e2 , e2 ’ e3 , . . . , en’1 ’ en , en }.

Dynkin diagram : c> c

c c ppp

1 2 n’1 n

Aut Dyn(¦) : {1}.

Dominant weights : Λ+ = { ai ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an ≥ 0 }.

1

Minimal weights : 0, 2 (e1 + e2 + · · · + en ).

Type Cn , n ≥ 1. Let V = Rn with canonical basis {ei }.

: ¦ = { ±2ei , ±ei ± ej | i > j }, 2n2 roots.

Root system

ai ei | ai ∈ Z,

Root lattice : Λr = { ai ∈ 2Z }.

: Λ = Zn , Z/2Z.

Weight lattice Λ/Λr

Simple roots : Π = {e1 ’ e2 , e2 ’ e3 , . . . , en’1 ’ en , 2en }.

Dynkin diagram : c< c

c c ppp

1 2 n’1 n

Aut Dyn(¦) : {1}.

Dominant weights : Λ+ = { ai ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an ≥ 0 }.

Minimal weights : 0, e1 .

§25. SPLIT SEMISIMPLE GROUPS 355

Type Dn , n ≥ 3. (For n = 2 the de¬nition works but yields A1 + A1 .) Let

V = Rn with canonical basis {ei }.

Root system : ¦ = { ±ei ± ej | i > j }, 2n(n ’ 1) roots.

ai ei | ai ∈ Z,

Root lattice : Λr = { ai ∈ 2Z }.

: Λ = Zn + 1 (e1 + e2 + · · · + en )Z,

Weight lattice 2

Z/2Z • Z/2Z if n is even,

Λ/Λr

Z/4Z if n is odd.

Simple roots : Π = {e1 ’ e2 , . . . , en’1 ’ en , en’1 + en }.

c n’1

c

Dynkin diagram : c c ppp

n’2 d

d cn

1 2

Aut Dyn(¦) : S3 if n = 4, {1, „ } if n = 3 or n > 4.

Dominant weights : Λ+ = { ai ei ∈ Λ | a1 ≥ a2 ≥ · · · ≥ an , an’1 + an ≥ 0 }.

1

Minimal weights : 0, e1 , 2 (e1 + e2 + · · · + en’1 ± en ).

Exceptional types.

Z/3Z.

E6 : Aut Dyn(¦) = {1, „ }, Λ/Λr

c c c c c

c

Z/2Z.

E7 : Aut Dyn(¦) = {1}, Λ/Λr

c c c c c c

c

E8 : Aut Dyn(¦) = {1}, Λ/Λr = 0.

c c c c c c c

c

F4 : Aut Dyn(¦) = {1}, Λ/Λr = 0. c> c