Speci¬c notations for the groups Sim(V, b), Iso(V, b) and PSim(V, b) are used ac-

cording to the type of b. If b is symmetric nonalternating, we set

O(V, b) = Iso(V, b), GO(V, b) = Sim(V, b) and PGO(V, b) = PSim(V, b);

if b is alternating, we let

Sp(V, b) = Iso(V, b), GSp(V, b) = Sim(V, b) and PGSp(V, b) = PSim(V, b).

Note that condition (??), de¬ning a similitude of (V, b) with multiplier ±, can be

rephrased as follows, using the adjoint involution σb :

(12.2) σb (g) —¦ g = ±IdV .

By taking the determinant of both sides, we obtain (det g)2 = ±n where n = dim V .

It follows that the determinant of an isometry is ±1 and that, if n is even,

det g = ±µ(g)n/2 for g ∈ Sim(V, b).

A ¬rst di¬erence between the orthogonal case and the symplectic case shows up in

the following result:

(12.3) Proposition. If b is a nonsingular alternating bilinear form on a vector

space V of dimension n (necessarily even), then

det g = µ(g)n/2 for g ∈ GSp(V, b).

Proof : Let g ∈ GSp(V, b) and let G, B denote the matrices of g and b respectively

with respect to some arbitrary basis of V . The matrix B is alternating and we have

Gt BG = µ(g)B.

By taking the pfa¬an of both sides, we obtain, by known formulas for pfa¬ans (see

Artin [?, Theorem 3.28]; compare with (??)):

det G pf B = µ(g)n/2 pf B,

hence det g = µ(g)n/2 .

By contrast, if b is symmetric and char F = 2, every hyperplane re¬‚ection is an

isometry with determinant ’1 (see (??)), hence it satis¬es det g = ’µ(g)n/2 .

We set

O+ (V, b) = { g ∈ O(V, b) | det g = 1 }.

Of course, O+ (V, b) = O(V, b) if char F = 2.

Similarly, if dim V = n is even, we set

GO+ (V, b) = { g ∈ GO(V, b) | det g = µ(g)n/2 },

and

PGO+ (V, b) = GO+ (V, b)/F — .

The elements in GO+ (V, b), O+ (V, b) are called proper similitudes and proper isome-

tries respectively.

If dim V is odd, there is a close relationship between similitudes and isometries,

as the next proposition shows:

(12.4) Proposition. Suppose that (V, b) is a nonsingular symmetric bilinear space

of odd dimension over an arbitrary ¬eld F ; then

GO(V, b) = O+ (V, b) · F — O+ (V, b) — F — O+ (V, b).

PGO(V, b)

and

§12. GENERAL PROPERTIES 155

Proof : If g is a similitude of (V, b) with multiplier ± ∈ F — , then by taking the

b we get ± ∈ F —2 . If ± = ±2 ,

determinant of both sides of the isometry ± · b 1

’1

then ±1 g is an isometry. Moreover, after changing the sign of ±1 if necessary, we

may assume that det(±’1 g) = 1. The factorization g = (±’1 g) · ±1 shows that

1 1

GO(V, b) = O+ (V, b) · F — , and the other isomorphisms are clear.

Hermitian spaces. Suppose (V, h) is a nonsingular hermitian space over a

quadratic separable ¬eld extension K of F (with respect to the nontrivial automor-

phism of K/F ). A similitude of (V, h) is an invertible linear map g : V ’ V for

which there exists a constant ± ∈ F — , called the multiplier of g, such that

(12.5) h g(v), g(w) = ±h(v, w) for v, w ∈ V .

As in the case of bilinear spaces, we write Sim(V, h) for the group of similitudes of

(V, h); let

µ : Sim(V, h) ’ F —

be the group homomorphism which carries every similitude to its multiplier; write

Iso(V, h) for the kernel of µ, whose elements are called isometries, and let

PSim(V, h) = Sim(V, h)/K — .

We also use the following more speci¬c notation:

U(V, h) = Iso(V, h), GU(V, h) = Sim(V, h), PGU(V, h) = PSim(V, h).

Condition (??) can be rephrased as

σh (g) —¦ g = ±IdV .

By taking the determinant of both sides, we obtain

NK/F (det g) = µ(g)n , where n = dim V .

This relation shows that the determinant of every isometry has norm 1. Set

SU(V, h) = { g ∈ U(V, h) | det g = 1 }.

Quadratic spaces. Let (V, q) be a nonsingular quadratic space over an arbi-

trary ¬eld F . A similitude of (V, q) is an invertible linear map g : V ’ V for which

there exists a constant ± ∈ F — , called the multiplier of g, such that

q g(v) = ±q(v) for v ∈ V .

The groups Sim(V, q), Iso(V, q), PSim(V, q) and the group homomorphism

µ : Sim(V, q) ’ F —

are de¬ned as for nonsingular bilinear forms. We also use the notation

O(V, q) = Iso(V, q), GO(V, q) = Sim(V, q), PGO(V, q) = PSim(V, q).

It is clear from the de¬nitions that every similitude of (V, q) is also a similitude of

its polar bilinear space (V, bq ), with the same multiplier, hence

GO(V, bq ) if char F = 2,

GO(V, q) ‚ Sim(V, b) =

GSp(V, bq ) if char F = 2,

and the reverse inclusion also holds if char F = 2.

For the rest of this section, we assume therefore char F = 2. If dim V is odd,

the same arguments as in (??) yield:

156 III. SIMILITUDES

(12.6) Proposition. Suppose (V, q) is a nonsingular symmetric quadratic space

of odd dimension over a ¬eld F of characteristic 2; then

GO(V, q) = O(V, q) · F — O(V, q) — F — PGO(V, q) O(V, q).

and

We omit the proof, since it is exactly the same as for (??), using the determinant

of q de¬ned in (??).

If dim V is even, we may again distinguish proper and improper similitudes, as

we now show.

By using a basis of V , we may represent the quadratic map q by a quadratic

form, which we denote again q. Let M be a matrix such that

q(X) = X t · M · X.

Since q is nonsingular, the matrix W = M + M t is invertible. Let g be a similitude

of V with multiplier ±, and let G be its matrix with respect to the chosen basis of V .

The equation q(G · X) = ±q(X) shows that the matrices Gt M G and ±M represent

the same quadratic form. Therefore, Gt M G ’ ±M is an alternating matrix. Let

R ∈ Mn (K) be such that

Gt M G ’ ±M = R ’ Rt .

(12.7) Proposition. The element tr(±’1 W ’1 R) ∈ K depends only on the simil-

itude g, and not on the choice of basis of V nor on the choices of matrices M

and R. It equals 0 or 1.

Proof : With a di¬erent choice of basis of V , the matrix G is replaced by G =

P ’1 GP for some invertible matrix P ∈ GLn (K), and the matrix M is replaced by

t

a matrix M = P t M P + U ’ U t for some matrix U . Then W = M + M is related

to W by W = P t W P . Suppose R, R are matrices such that

t t

Gt M G ’ ±M = R ’ Rt

(12.8) and G M G ’ ±M = R ’ R .

By adding each side to its transpose, we derive from these equations:

t

Gt W G = ±W

(12.9) and G W G = ±W .

In order to prove that tr(±’1 W ’1 R) depends only on the similitude g, we have

’1

to show tr(W ’1 R) = tr(W R ). By substituting for M its expression in terms

t t

of M , we derive from (??) that R ’ R = R ’ R , where

R = P t RP + P t Gt (P ’1 )t U P ’1 GP ’ ±U,

(12.10)

’1

hence R = R + S for some symmetric matrix S ∈ Mn (K). Since W =

’1 ’1 ’1 ’1 t ’1

W MW + (W M W ) , it follows that W is alternating. By (??),

alternating matrices are orthogonal to symmetric matrices for the trace bilinear

’1 ’1

form, hence tr(g R ) = tr(g R ). In view of (??) we have

’1

R ) = tr(P ’1 W ’1 RP ) + tr P ’1 W ’1 Gt (P ’1 )t U P ’1 GP

(12.11) tr(W

+ ± tr P ’1 W ’1 (P ’1 )t U .

By (??), W ’1 Gt = ±G’1 W ’1 , hence the second term on the right side of (??)

equals

± tr P ’1 G’1 W ’1 (P ’1 )t U P ’1 GP = ± tr W ’1 (P ’1 )t U P ’1 .

§12. GENERAL PROPERTIES 157

Therefore, the last two terms on the right side of (??) cancel, and we get

’1 ’1

R ) = tr(W ’1 R),

tr(W R ) = tr(W

proving that tr(±’1 W ’1 R) depends only on the similitude g.

In order to prove that this element is 0 or 1, we compute s2 (W ’1 M ), the coef-

¬cient of X n’2 in the characteristic polynomial of W ’1 M (see (??)). By (??), we

have G’1 W ’1 = ±’1 W ’1 Gt , hence G’1 W ’1 M G = ±’1 W ’1 Gt M G, and there-

fore

s2 (W ’1 M ) = s2 (±’1 W ’1 Gt M G).

On the other hand, (??) also yields Gt M G = ±M + R ’ Rt , hence by substituting

this in the right side of the preceding equation we get

s2 (W ’1 M ) = s2 (W ’1 M + ±’1 W ’1 R ’ ±’1 W ’1 Rt ).

By (??), we may expand the right side to get

s2 (W ’1 M ) = s2 (W ’1 M ) + tr(±’1 W ’1 R) + tr(±’1 W ’1 R)2 .

Therefore, tr(±’1 W ’1 R) + tr(±’1 W ’1 R)2 = 0, hence

tr(±’1 W ’1 R) = 0, 1.

(12.12) De¬nition. Let (V, q) be a nonsingular quadratic space of even dimension

over a ¬eld F of characteristic 2. Keep the same notation as above. In view of the

preceding proposition, we set

∆(g) = tr(±’1 W ’1 R) ∈ {0, 1} for g ∈ GO(V, q).

Straightforward veri¬cations show that ∆ is a group homomorphism

∆ : GO(V, q) ’ Z/2Z,

called the Dickson invariant. We write GO+ (V, q) for the kernel of this homomor-

phism. Its elements are called proper similitudes, and the similitudes which are

mapped to 1 under ∆ are called improper. We also let

O+ (V, q) = { g ∈ O(V, q) | ∆(g) = 0 } and PGO+ (V, q) = GO+ (V, q)/F — .

(12.13) Example. Let dim V = n = 2m. For any anisotropic vector v ∈ V , the

hyperplane re¬‚ection ρv : V ’ V is de¬ned in arbitrary characteristic by

ρv (x) = x ’ vq(v)’1 bq (v, x) for x ∈ V .

This map is an isometry of (V, q). We claim that it is improper.

This is clear if char F = 2, since the matrix of ρv with respect to an orthogonal

basis whose ¬rst vector is v is diagonal with diagonal entries (’1, 1, . . . , 1), hence

det ρv = ’1.

If char F = 2, we compute ∆(ρv ) by means of a symplectic basis (e1 , . . . , en )

of (V, bq ) such that e1 = v. With respect to that basis, the quadratic form q is

represented by the matrix

«

M1 0

q(e2i’1 ) 0

¬ ·

..

M = where Mi = ,

. 1 q(e2i )

0 Mm

158 III. SIMILITUDES

and the map ρv is represented by

«

G1 0

1 q(e1 )’1

¬ ·

..

G= where G1 = , Gi = I, i ≥ 2.

. 0 1

0 Gm

As a matrix R such that Gt M G + M = R + Rt we may take

«

R1 0

01

¬ ·

..

R= where R1 = , Ri = I, i ≥ 2.

. 00

0 Rm

It is readily veri¬ed that tr(W ’1 R) = 1, hence ∆(ρv ) = 1, proving the claim.

12.B. Similitudes of algebras with involution. In view of the charac-

terization of similitudes of bilinear or hermitian spaces by means of the adjoint

involution (see (??)), the following de¬nition is natural:

(12.14) De¬nition. Let (A, σ) be a central simple F -algebra with involution. A

similitude of (A, σ) is an element g ∈ A such that

σ(g)g ∈ F — .

The scalar σ(g)g is called the multiplier of g and is denoted µ(g). The set of all

similitudes of (A, σ) is a subgroup of A— which we call Sim(A, σ), and the map µ

is a group homomorphism

µ : Sim(A, σ) ’ F — .

It is then clear that similitudes of bilinear spaces are similitudes of their endo-

morphism algebras:

Sim EndF (V ), σb = Sim(V, b)

if (V, b) is a nonsingular symmetric or alternating bilinear space. There is a corre-

sponding result for hermitian spaces.

Similitudes can also be characterized in terms of automorphisms of the algebra

with involution. Recall that an automorphism of (A, σ) is an F -algebra automor-

phism which commutes with σ:

AutF (A, σ) = { θ ∈ AutF (A) | σ —¦ θ = θ —¦ σ }.

Let K be the center of A, so that K = F if σ is of the ¬rst kind and K is a quadratic

´tale F -algebra if σ is of the second kind. De¬ne Aut K (A, σ) = AutF (A, σ) ©

e

AutK (A).

(12.15) Theorem. With the notation above,

AutK (A, σ) = { Int(g) | g ∈ Sim(A, σ) }.

There is therefore an exact sequence:

Int

1 ’ K — ’ Sim(A, σ) ’’ AutK (A, σ) ’ 1.

’

Proof : By the Skolem-Noether theorem, every automorphism of A over K has the

form Int(g) for some g ∈ A— . Since

σ —¦ Int(g) = Int σ(g)’1 —¦ σ,

§12. GENERAL PROPERTIES 159

the automorphism Int(g) commutes with σ if and only if σ(g)’1 ≡ g mod K — , i.e.,

σ(g)g ∈ K — . Since σ(g)g is invariant under σ, the latter condition is also equivalent

to σ(g)g ∈ F — .

Let PSim(A, σ) be the group of projective similitudes, de¬ned as

PSim(A, σ) = Sim(A, σ)/K — .

In view of the preceding theorem, the map Int de¬nes a natural isomorphism

∼

PSim(A, σ) ’ AutK (A, σ).

’

Speci¬c notations for the groups Sim(A, σ) and PSim(A, σ) are used according

to the type of σ, re¬‚ecting the notations for similitudes of bilinear or hermitian

spaces:

±

GO(A, σ) if σ is of orthogonal type,

Sim(A, σ) = GSp(A, σ) if σ is of symplectic type,

GU(A, σ) if σ is of unitary type,

and

±

PGO(A, σ) if σ is of orthogonal type,

PSim(A, σ) = PGSp(A, σ) if σ is of symplectic type,

PGU(A, σ) if σ is of unitary type.

Similitudes with multiplier 1 are isometries; they make up the group Iso(A, σ):

Iso(A, σ) = { g ∈ A— | σ(g) = g ’1 }.

We also use the following notation:

±

O(A, σ) if σ is of orthogonal type,

Iso(A, σ) = Sp(A, σ) if σ is of symplectic type,

U(A, σ) if σ is of unitary type.

For quadratic pairs, the corresponding notions are de¬ned as follows:

(12.16) De¬nition. Let (σ, f ) be a quadratic pair on a central simple F -algebra A.

An automorphism of (A, σ, f ) is an F -algebra automorphism θ of A such that

σ—¦θ =θ—¦σ and f —¦ θ = f.

A similitude of (A, σ, f ) is an element g ∈ A— such that σ(g)g ∈ F — and f (gsg ’1 ) =

f (s) for all s ∈ Sym(A, σ). Let GO(A, σ, f ) be the group of similitudes of (A, σ, f ),

let

PGO(A, σ, f ) = GO(A, σ, f )/F —

and write AutF (A, σ, f ) for the group of automorphisms of (A, σ, f ). The same

arguments as in (??) yield an exact sequence

Int

1 ’ F — ’ GO(A, σ, f ) ’’ AutF (A, σ, f ) ’ 1,

’

hence also an isomorphism

∼

PGO(A, σ, f ) ’ AutF (A, σ, f ).

’

160 III. SIMILITUDES

For g ∈ GO(A, σ, f ) we set µ(g) = σ(g)g ∈ F — . The element µ(g) is called the

multiplier of g and the map

µ : GO(A, σ, f ) ’ F —

is a group homomorphism. Its kernel is denoted O(A, σ, f ).

It is clear from the de¬nition that GO(A, σ, f ) ‚ Sim(A, σ). If char F = 2, the

map f is the restriction of 1 TrdA to Sym(A, σ), hence the condition f (gsg ’1 ) =

2

f (s) for all s ∈ Sym(A, σ) holds for all g ∈ GO(A, σ). Therefore, we have in this

case

GO(A, σ, f ) = GO(A, σ), PGO(A, σ, f ) = PGO(A, σ) and O(A, σ, f ) = O(A, σ).

In particular, if (V, q) is a nonsingular quadratic space over F and (σq , fq ) is the

associated quadratic pair on EndF (V ) (see (??)),

GO EndF (V ), σq , fq = GO EndF (V ), σq = GO(V, q).

There is a corresponding result if char F = 2:

(12.17) Example. Let (V, q) be a nonsingular quadratic space of even dimension

over a ¬eld F of characteristic 2, and let (σq , fq ) be the associated quadratic pair

on EndF (V ). We claim that

GO EndF (V ), σq , fq = GO(V, q),

hence also PGO EndF (V ), σq , fq = PGO(V, q) and O EndF (V ), σq , fq = O(V, q).

In order to prove these equalities, observe ¬rst that the standard identi¬cation

•q of (??) associated with the polar of q satis¬es the following property: for all

g ∈ EndF (V ), and for all v, w ∈ V ,

g —¦ •q (v — w) —¦ σq (g) = •q g(v) — g(w) .

Therefore, if g ∈ GO EndF (V, σq , fq ) and ± = µ(g) ∈ F — , the condition

fq g —¦ •q (v — v) —¦ g ’1 = fq —¦ •q (v — v) for v ∈ V

amounts to

q g(v) = ±q(v) for v ∈ V ,

which means that g is a similitude of the quadratic space (V, q), with multiplier ±.

This shows GO EndF (V ), σq , fq ‚ GO(V, q).

For the reverse inclusion, observe that if g is a similitude of (V, q) with multiplier

±, then σq (g)g = ± since g also is a similitude of the associated bilinear space (V, bq ).

Moreover, the same calculation as above shows that

fq g —¦ •q (v — v) —¦ g ’1 = fq —¦ •q (v — v) for v ∈ V .

Since Sym EndF (V ), σq , fq is spanned by elements of the form •q (v —v), it follows

that fq (gsg ’1 ) = fq (s) for all s ∈ Sym(A, σ), hence g ∈ GO EndF (V ), σq , fq . This

proves the claim.

We next determine the groups of similitudes for quaternion algebras.

(12.18) Example. Let Q be a quaternion algebra with canonical (symplectic)

involution γ over an arbitrary ¬eld F . Since γ(q)q ∈ F for all q ∈ Q, we have

Sim(Q, γ) = GSp(Q, γ) = Q— .

Therefore, γ commutes with all the inner automorphisms of Q. (This observation

also follows from the fact that γ is the unique symplectic involution of Q: for

§12. GENERAL PROPERTIES 161

every automorphism θ, the composite θ —¦ γ —¦ θ ’1 is a symplectic involution, hence

θ —¦ γ —¦ θ’1 = γ).

Let σ be an orthogonal involution on Q; by (??) we have

σ = Int(u) —¦ γ

for some invertible quaternion u such that γ(u) = ’u and u ∈ F . Since γ commutes

with all automorphisms of Q, an inner automorphism Int(g) commutes with σ if

and only if it commutes with Int(u), i.e., gu ≡ ug mod F — . If » ∈ F — is such that

gu = »ug, then by taking the reduced norm of both sides of this equation we obtain

»2 = 1, hence gu = ±ug. The group of similitudes of (Q, σ) therefore consists of

the invertible elements which commute or anticommute with u. If char F = 2, we

thus obtain

GO(Q, σ) = F (u)— .

If char F = 2, let v be any invertible element which anticommutes with u; then

GO(Q, σ) = F (u)— ∪ F (u)— · v .

Finally, we consider the case of quadratic pairs on Q. We assume that char F =

2 since, if the characteristic is di¬erent from 2, the similitudes of a quadratic pair

(σ, f ) are exactly the similitudes of the orthogonal involution σ. Since char F =

2, every involution which is part of a quadratic pair is symplectic, hence every

quadratic pair on Q has the form (γ, f ) for some linear map f : Sym(Q, γ) ’ F .

Take any ∈ Q satisfying

f (s) = TrdQ ( s) for s ∈ Sym(Q, γ)

(see (??)). The element is uniquely determined by the quadratic pair (γ, f ) up to

the addition of an element in Alt(Q, γ) = F , and it satis¬es TrdQ ( ) = 1, by (??)

and (??). Therefore, there exists an element v ∈ Q— such that v ’1 v = + 1. We

claim that

GO(Q, γ, f ) = F ( )— ∪ F ( )— · v .

Since GSp(Q, γ) = Q— , an element g ∈ Q— is a similitude of (Q, γ, f ) if and only

if f (gsg ’1 ) = f (s) for all s ∈ Sym(Q, γ). By de¬nition of , this condition can be

rephrased as

TrdQ ( gsg ’1 ) = TrdQ ( s) for s ∈ Sym(Q, γ).

Since the left-hand expression equals TrdQ (g ’1 gs), this condition is also equivalent

to

TrdQ ( ’ g ’1 g)s = 0 for s ∈ Sym(Q, γ);

that is, ’ g ’1 g ∈ F , since F = Alt(Q, γ) is the orthogonal space of Sym(Q, γ)

for the trace bilinear form (see (??)). Suppose that this condition holds and let

» = ’ g ’1 g ∈ F . We proceed to show that » = 0 or 1. Since TrdQ ( ) = 1 we

have NrdQ ( ) = 2 + and NrdQ ( + ») = 2 + + »2 + ». On the other hand, we

must have NrdQ ( ) = NrdQ ( + »), since + » = g ’1 g. Therefore, »2 + » = 0 and

» = 0 or 1. Therefore,

GO(Q, γ, f ) = { g ∈ Q— | g ’1 g = } ∪ { g ∈ Q— | g ’1 g = + 1 },

and the claim is proved.

162 III. SIMILITUDES

(12.19) Example. Let A = Q1 —F Q2 be a tensor product of two quaternion

algebras over a ¬eld F of characteristic di¬erent from 2, and σ = γ1 — γ2 , the

tensor product of the canonical involutions. A direct computation shows that the

Lie algebra Alt(A, σ) decomposes as a direct sum of the (Lie) algebras of pure

quaternions in Q1 and Q2 :

Alt(A, σ) = (Q0 — 1) • (1 — Q0 ).

1 2

Since Lie algebras of pure quaternions are simple and since the decomposition of