Some twisted Groups - Lectures on Chevalley groups

Lectures on Chevalley groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 13 July 2013

§11. Some twisted Groups

In this section we study the group $G_{σ}$ of fixed points of a Chevalley group $G$ under an automorphism $σ .$ We consider only the simplest case, in which $σ$ fixes $U, H, U^{-}, N,$ hence acts on $W = N / H$ and permutes the $𝔛_{α}'s.$ Before launching into the general theory, we consider some examples:

(a) $G = {S L}_{n} .$ If $σ$ is a nontrivial graph automorphism, it has the form $σ x = a {x'}^{- 1} a^{- 1}$ (where $x'$ is the transpose of $x$ and

a = [\begin{matrix} ε_{1} \\ ε_{2} \\ ⋰ \end{matrix}],

$ε_{i} = \pm 1).$ We see that $σ$ fixes $x$ if and only if $x a x' = a .$ If $a$ is skew, we get $G_{σ} = {S p}_{n} .$ If $a$ is symmetric, we get $G_{σ} = {S O}_{n}$ (split form). The group ${S O}_{2 n}$ in characteristic 2 does not arise here, but it can be recovered as a subgroup of ${S O}_{2 n + 1},$ namely the one "supported" by the long roots.

Let $t \to \overline{t}$ be an involutory automorphism of $k$ having $k_{0}$ as fixed field. If $σ$ is now modified so that $σ x = a {\overline{x'}}^{- 1} a^{- 1},$ then $G_{σ} = {S U}_{n}$ (split form). This last result holds even if $k$ is a division ring provided $t \to \overline{t}$ is an anti-automorphism.

If $V$ is the vector space over $ℝ$ generated by the roots and $W$ is the Weyl group, then $σ$ acts on $V$ and $W$ and has fixed point subspaces $V_{σ}$ and $W_{σ} .$ $W_{σ}$ is a reflection group on $V_{σ}$ with the corresponding "roots" being the projection on $V_{σ}$ of the original roots. To see these facts, we write $n = 2 m + 1$ or $n = 2 m$ and use the indices $- m,$ $- (m - 1),$ $\dots,$ $m - 1,$ $m$ with the index $0$ omitted in case $n = 2 m .$ If $ω_{i}$ is the weight on $H$ defined by $ω_{i} : diag (a_{- m}, \dots, a_{m}) \to a_{i},$ then the roots are $ω_{i} - ω_{j}$ $(i \neq j)$ and $σ ω_{i} = - ω_{- i} .$ $V_{σ}$ is thus spanned by ${ω_{i}^{'} = ω_{- i} - ω_{i} | i > 0} .$ Now $w \in W_{σ}$ if and only if $w$ commutes with $σ,$ i.e., if and only if $w (ω_{i} - ω_{j}) = ω_{k} - ω_{ℓ}$ implies $w (ω_{- i} - ω_{- j}) = ω_{- k} - ω_{- ℓ} .$ We see that $W_{σ}$ is the octahedral group acting on $V_{σ}$ by all permutations and sign changes of the basis ${ω_{i}^{'}} .$ The projection of $ω_{i} - ω_{j}$ $(i, h \neq 0)$ on $V_{σ}$ is $\frac{1}{2} (ω_{i} - ω_{- i} - ω_{j} + ω_{- j}) = \frac{1}{2} (\pm ω_{k}^{'} \pm ω_{ℓ}^{'})$ $(k, ℓ > 0)$ or $\pm ω_{k}^{'}$ $(k > 0) .$ If either $i = 0$ or $j = 0$ the projection of $ω_{i} - ω_{j}$ is $\pm \frac{1}{2} ω_{k}^{'}$ $(k > 0) .$ The projected system is of type $C_{m}$ if $n = 2 m$ or $B C_{m}$ (a combination of $B_{m}$ and $C_{m})$ if $n = 2 m + 1 .$

(b) $G = {S O}_{2 n}$ (split form, char $k \neq 2).$ We take the group defined by the form $f = 2 Σ_{i = 1}^{n} x_{i} x_{- i} .$ We will take the graph automorphism to be $σ x = a_{1} a {x'}^{- 1} a^{- 1} a_{1}^{- 1}$

(a = [\begin{matrix} 1 \\ ⋰ \\ 1 \end{matrix}], a_{1} = [\begin{matrix} 1 \\ ⋱ \\ 1 \\ 0 & 1 \\ 1 & 0 \\ 1 \\ ⋱ \\ 1 \end{matrix}]) .

The corresponding form fixed by elements of $G_{σ}$ is $f' = 2 Σ_{i = 2}^{n} x_{i} x_{- i} + x_{1}^{2} + x_{- 1}^{2} .$ Thus, $G_{σ}$ fixes $f' - f = {(x_{1} - x_{- 1})}^{2}$ and hence the hyperplane $x_{1} - x_{- 1} = 0 .$ $G_{σ}$ on this hyperplane is the group ${S O}_{2 n - 1} .$

If we now combine $σ$ with $t \to \overline{t}$ as in (a), the form $f'$ is replaced by $f^{''} = Σ_{i = 2}^{n} (x_{i} {\overline{x}}_{- i} + x_{- i} {\overline{x}}_{i}) + x_{1} {\overline{x}}_{1} + x_{- 1} {\overline{x}}_{- 1} .$ If we make the change of coordinates $x_{1}$ replaced by $x_{1} + t x_{- 1} x_{- 1}$ replaced by $x_{1} + \overline{t} x_{- 1}$ $(t \in k, t \neq \overline{t}),$ we see that $f$ is replaced by $2 Σ_{i = 2}^{n} x_{i} x_{- i} + 2 (x_{1}^{2} + a x_{1} x_{- 1} + b x_{- 1}^{2})$ and $f^{''}$ is replaced by $Σ_{i = 2}^{n} (x_{i} {\overline{x}}_{- i} + x_{- i} {\overline{x}}_{i}) + (2 x_{1} {\overline{x}}_{1} + a (x_{1} {\overline{x}}_{- 1} + x_{- 1} {\overline{x}}_{1}) + 2 b x_{- 1} {\overline{x}}_{- 1}),$ where $a = t + \overline{t}$ and $b = t \overline{t} .$ Since these two forms have the same matrix, $G_{σ}$ is ${S O}_{2 n}$ over $k_{0}$ re the new version of $f .$ That is, $G_{σ}$ is ${S O}_{2 n} (k_{0})$ for a form of index $n - 1$ which has index $n$ over $k .$

Example: If $n = 4,$ $k = ℂ,$ and $k_{0} = ℝ, G_{σ}$ is the Lorentz group (re $f = x_{1}^{2} - x_{2}^{2} - x_{3}^{2} - x_{4}^{2}).$ If we observe that $D_{2}$ corresponds to $A_{1} \times A_{1},$ we see that ${S L}_{2} (ℂ)$ and the $0 -component$ of the Lorentz group are isomorphic over their centers. Thus, ${S L}_{2} (ℂ)$ is the universal covering group of the connected Lorentz group.

Exercise: Work out $D_{3} \sim A_{3}$ in the same way.

For other examples see E. Cartan, Oeuvres Complètes, No. 38, especially at the end.

Aside from the specific facts worked out in the above examples we should note the following. In the single root length case, the fixed point set of a graph automorphism yields no new group, only an imbedding of one Chevalley group in another (e.g. ${S p}_{n}$ or ${S O}_{n}$ in ${S L}_{n}).$ To get a new group (e.g. ${S U}_{n})$ we must use a field automorphism as well.

Now to start our general development we will consider first the effect of twisting abstract reflection groups and root systems. Let $V$ be a finite dimensional real Euclidean vector space and let $Σ$ be a finite set of nonzero elements of $V$ satisfying

(1)	$α \in Σ$ implies $c α \notin Σ$ if $c > 0,$ $c \neq 1 .$
(2)	$w_{α} Σ = Σ$ for all $α \in Σ$ where $w_{α}$ is the reflection in the hyperplane orthogonal to $α .$

(See Appendix I). We pick an ordering on

V

and let

P

(respectively

Π)

be the positive (respectively simple) elements of

Σ

relative to that ordering. Suppose

σ

is an automorphism of

V

which permutes the positive multiples of the elements of each of

Σ,

P,

and

Π .

It is not required that

σ

fix

Σ,

although it will if all elements of

Σ

have the same length. Let

ρ

be the corresponding permutation of the roots. Note that

σ

is of finite order and normalizes

W .

Let

V_{σ}

and

W_{σ}

denote the fixed points in

V

and

W

respectively. If

\overline{α}

is the average of the elements in the

σ -orbit

α,

then

(β, \overline{α}) = (β, α)

for all

β \in V_{σ} .

Hence the projection of

α

V_{σ}

\overline{α} .

Theorem 32: Let $Σ, P, Π, σ$ etc. be as above.

(a)	The restriction of $W_{σ}$ to $V_{σ}$ is faithful.
(b)	$W_{σ} \| V_{σ}$ is a reflection group.
(c)	If $Σ_{σ}$ denotes the projection of $Σ$ on $V_{σ},$ then $Σ_{σ}$ is the corresponding "root system"; i.e., ${w_{\overline{α}} \| V_{σ}, \overline{α} \in Σ_{σ}}$ generates $W_{σ} \| V_{σ}$ and $w_{\overline{α}} Σ_{σ} = Σ_{σ} .$ However, (1) may fail for $Σ_{σ} .$
(d)	If $Π_{σ}$ is the projection of $Π$ on $V_{σ},$ then $Π_{σ}$ is the corresponding "simple system"; i.e. if multiples are cast out (in case (1) fails for $Π_{σ}),$ then $Π_{σ}$ is linearly independent and the positive elements of $Σ_{σ}$ are positive linear combinations of elements of $Π_{σ} .$

Proof.

Denote the projection of $V$ on $V_{σ}$ by $v \to \overline{v} .$ This commutes with $σ$ and with all elements of $W_{σ} .$

(1) If $α \in Σ,$ then $\overline{α} \neq 0;$ indeed $α > 0$ implies $\overline{α} > 0 .$ If $α$ is positive, so are all vectors in the $σ -orbit$ of $α .$ Thus, their average $\overline{α}$ is also positive. If $α < 0,$ then $\overline{α} = - (- \overline{α}) < 0 .$

(2) Proof of (a). If $w \in W_{σ},$ $w \neq 1,$ then $w α < 0$ for some root $α > 0 .$ Thus, $w \overline{α} = \overline{w α} < 0$ and $\overline{α} > 0 .$ So $w | V_{σ} \neq 1 .$

(3) Let $π$ be a $ρ -orbit$ of simple roots, let $W_{π}$ be the group generated by all $w_{α}$ $(α \in π),$ let $P_{π}$ be the corresponding set of positive roots, and let $w_{π}$ be the unique element of $W_{π}$ so that $w_{π} P_{π} = - P_{π} .$ Then $w_{π} \in W_{σ}$ and $w_{π} | V_{σ} = w_{\overline{α}} | V_{σ}$ for any root $α \in P_{π} .$ To see this, first consider $σ w_{π} σ^{- 1} \in W_{π} .$ Since $σ w_{π} σ^{- 1} P_{π} = P_{π},$ then $σ w_{π} σ^{- 1} = w_{π}$ by uniqueness, and $w_{π} \in W_{σ} .$ Since $ρ$ permutes the elements of $π$ in a single orbit, the projections on $V_{σ}$ of the elements of $P_{π}$ are all positive multiples of each other. It follows that if $α$ is any element of $P_{π},$ then $w_{π} \overline{α} = - \overline{α} .$ If $v \in V_{σ}$ with $(v, \overline{α}) = 0,$ then $0 = (v, \overline{β}) = (v, β)$ for $β \in π .$ Hence $w_{π} v = v .$ Thus $w_{π} | V_{σ} = w_{\overline{α}} | V_{σ} .$

(4) If $ν$ is a $ρ -orbit$ of roots and $w \in W_{σ}$ then all elements of $w ν$ have the same sign. This follows from $w σ α = σ w α$ for $α \in Σ,$ $w \in W_{σ} .$

(5) ${w_{π} | π$ a $ρ -orbit$ of simple $roots}$ generates $W_{σ} .$ Let $w \in W_{σ}$ with $w \neq 1$ and let $α$ be a simple root such that $w α < 0 .$ Let $π$ be the $ρ -orbit$ containing $α .$ By (4), $w P_{π} < 0$ (i.e., $w β < 0$ for all $β \in P_{π}).$ Now $w w_{π} P_{π} > 0$ and $w_{π}$ permutes the elements of $P - P_{π} .$ Hence, $N (w w_{π}) = N (w) - N (w_{π})$ (see Appendix 11.17). Using induction on $N (w),$ we may thus show that $w$ is a product of $w_{π}'s.$

(6) If $w_{0}$ is the element of $W$ such that $w_{0} P = - P,$ then $w_{0} \in W_{σ} .$ This follows from $σ w_{0} σ^{- 1} P = - P$ and the uniqueness of $w_{0} .$

(7) ${w P_{π} | w \in W_{σ}, π$ a $ρ -orbit$ of simple $roots}$ is a partition of $Σ .$ If the $w P_{π}'s$ are called parts, then $α, β$ belong to the same part if and only if $\overline{α} = c \overline{β}$ for some $c > 0 .$ To prove (7), we consider $α \in Σ,$ $α > 0 .$ Now $w_{0} α < 0$ and $w_{0} = w_{1} w_{2} \dots w_{r}$ where each $w_{i} = w_{π}$ for some $ρ -orbit$ of simple roots $π$ (by (5) and (6)). Choose $i$ so that $w_{i + 1} \dots w_{r} α > 0$ and $w_{i} w_{i + 1} \dots w_{r} α < 0 .$ If $w_{i} = w_{π},$ then $w_{i + 1} \dots w_{r} α \in P_{π};$ i.e., $α$ is in some part. Similarly, if $α < 0,$ $α$ is in some part. Now assume $α, β$ belong to the same part, say to $w P_{π} .$ We may assume $α, β \in P_{π} .$ Then $\overline{α}$ and $\overline{β}$ are positive multiples of each other, as has been noted in (3). Conversely, assume

8) $Σ_{0}$ consists of all $w \overline{α}$ such that $w \in W_{0}$ and $α$ is a root whose support lies in a simple $ρ -orbit.$ Now $\overline{β}$ has its support in $π$ and hence so does $β$ since $σ$ maps simple roots not in $π$ to positive multiples of simple roots not in $π .$ We see then that $β \in P_{π},$ and that any part containing $α$ also contains $β .$ The parts are just the sets of $β$ such that $\overline{β} = c \overline{α},$ $c > 0$ and hence form a partition.

(8) ${w \overline{α} | w \in W_{σ}, α$ has support in a $ρ -orbit$ of simple $roots} = Σ_{σ} .$

(9) Parts (b) and (c) follow from (3), (5), and (8).

(10) Proof of (d). We select one root $α$ from each $ρ -orbit$ and form ${α} .$ This set, consisting of elements whose supports in $Π$ are disjoint, is independent since $Π$ is. If $α > 0$ then it is a positive linear combination of the elements of $Π .$ Hence $\overline{α}$ is a positive linear combination of the elements of $Π_{σ} .$

$□$

Remark: To achieve condition (1) for a root system, we can stick to the set of shortest projections in the various directions.

Examples:

(a)	For $σ$ of order 2, $W$ of type $A_{2 n - 1},$ we get $W_{σ}$ of type $C_{n} .$ For $W$ of type $A_{2 n},$ we get $W_{σ}$ of type $B C_{n} .$
(b)	For $σ$ of order 2, $W$ of type $D_{n},$ we get $W_{σ}$ of type $B_{n - 1} .$
(c)	For $σ$ of order 3, $W$ of type $D_{4},$ we get $W_{σ}$ of type $G_{2} .$ To see this let $α, β, γ, δ$ be the simple roots with $δ$ connected with $α, β,$ and $γ .$ Then $\overline{α} = \frac{1}{3} (α + β + γ),$ $\overline{δ} = δ$ and $⟨ \overline{α}, \overline{δ} ⟩ = - 1,$ $⟨ \overline{δ}, α ⟩ = - 3,$ giving $W_{σ}$ of type $G_{2} .$ Schematically: $\begin{matrix} = & ⟶ \end{matrix}$
(d)	For $σ$ of order 2, $W$ of type $E_{6},$ we get $W_{σ}$ of type $F_{4} .$ $\begin{matrix} = & ⟶ \end{matrix}$
(e)	For $σ$ or order 2, $W$ of type $C_{2},$ we get $W_{σ}$ of type $A_{1} .$
(f)	For $σ$ or order 2, $W$ of type $G_{2},$ we get $W_{σ}$ of type $A_{1} .$
(g)	For $σ$ or order 2, $W$ of type $F_{4},$ we get $W_{σ}$ of type $𝒟_{16}$ (the dihedral group of order 16). To see this let $\begin{matrix} \end{matrix}$ be the Dynkin diagram of $F_{4},$ and $σ α = \sqrt{2} δ,$ $σ β = \sqrt{2} γ .$ Since $\overline{α} = \frac{1}{2} (α + \sqrt{2} δ),$ $\overline{β} = \frac{1}{2} (β + \sqrt{2} γ),$ we have $⟨ \overline{β}, \overline{α} ⟩ = - 1, ⟨ \overline{α}, \overline{β} ⟩ = - (2 + \sqrt{2}) .$ This corresponds to an angle of $7 π / 8$ between $\overline{α}$ and $\overline{β} .$ Hence $W_{σ}$ is of type $𝒟_{16} .$ Alternatively, we note that $w_{\overline{α}} w_{\overline{β}}$ makes six positive roots negative and that there are 24 positive roots in all, so that $w_{0} = {(w_{\overline{α}} w_{\overline{β}})}^{4} .$ Hence, $w_{\overline{α}}^{2} = w_{\overline{β}}^{2} = {(w_{\overline{α}} w_{\overline{β}})}^{8} = 1$ and $W_{σ}$ is of type $𝒟_{16} .$ Note that this is the only case of those we have considered in which $W_{σ}$ fails to be crystallographic (See Appendix V).

In (e), (f), (g) we are assuming that multiples have been cast out.

The partition of $Σ$ in (7) above can be used to define an equivalence relation $R$ on $Σ$ by $α \equiv β$ if and only if $\overline{α}$ is a positive multiple of $\overline{β}$ where $\overline{α}$ is the projection of $α$ on $V_{σ} .$ Letting $Σ / R$ denote the collection of equivalence classes we have the following:

Corollary: If $Σ$ is crystallographic and indecomposable, then an element of $Σ / R$ is the positive system of roots of a system of one of the following types:

(a)	$A_{1}^{n}$ $n = 1, 2,$ or $3 .$
(b)	$A_{2}$ (this occurs only if $Σ$ is of type $A_{2 n}).$
(c)	$C_{2}$ (this occurs if $Σ$ is of type $C_{2}$ or $F_{4}).$
(d)	$G_{2} .$

Now let $G$ be a Chevalley group over a field $k$ of characteristic $p .$ Let $σ$ be an automorphism of $G$ which is the product of a graph automorphism and a field automorphism $θ$ of $k$ and such that if $ρ$ is the corresponding permutation of the roots then

(1)	if $ρ$ preserves lengths, then order $θ =$ order $ρ .$
(2)	if $ρ$ doesn't preserve lengths, then $p θ^{2} = 1$ (where $p$ is the map $x \to x^{p}).$

(Condition (1) focuses our attention on the only interesting case. Observe that

ρ = id.,

θ = id.

is allowed. Condition (2) could be replaced by

θ^{2} = p

thereby extending the development to follow, suitably modified, to imperfect fields

k .)

We know that

p = 2

G

is of type

C_{2}

F_{4}

and

p = 3

G

is of type

G_{2} .

Recall also that

σ x_{α} (t) = {\begin{matrix} x_{ρ α} (ε_{α} t^{θ}) & if | α | \geq | ρ α | \\ x_{ρ α} (ε_{α} t^{p θ}) & if | α | < | ρ α | \end{matrix} where

$ε_{α} = \pm 1$ and $ε_{α} = 1$ if $\pm α$ is simple. (See the proof of Theorem 29.)

Now $σ$ preserves $U, H, B, U^{-},$ and $N,$ and hence $N / H ≅ W .$ The action thus induced on $W$ is concordant with the permutation $ρ$ of the roots. Since $ρ$ preserves angles, it agrees up to positive multiples with an isometry on the real space generated by the roots. Thus the results of Theorem 32 may be applied. Also we observe that if $n$ is the order of $ρ,$ then $n = 1, 2,$ or $3,$ so that the length of each $ρ -orbit$ is $1$ or $n .$

Lemma 60: $Π ε_{α} = 1$ over each $ρ -orbit$ of length $n .$

Proof.

Since $σ^{n}$ acts on each $𝔛_{α}$ $(\pm α$ simple) as a field automorphism, it does so on all of $G,$ whence the lemma.

$□$

Lemma 61: If $α \in Σ / R,$ then $𝔛_{α, σ} \neq 1 .$

Proof.

Choose $α \in a$ so that no $β \in a$ can be added to it to yield another root. If the orbit of $α$ has length 1, set $x = x_{α} (1)$ if $ε_{α} = 1,$ $x = x_{α} (t)$ with $t \in k,$ $t \neq 0$ and $t + t^{θ} = 0$ if $ε_{α} \neq 1 .$ Then $x \in 𝔛_{α, σ} .$ If the length is $n,$ we set $y = x_{α} (1),$ then $x = y \cdot σ y \cdot σ^{2} y \dots$ over the orbit, and use Lemma 60.

$□$

Theorem 33: Let $G, σ,$ etc. be as above.

(a)	For each $w \in W_{σ},$ the group $U_{w} = U \cap w^{- 1} U^{-} w$ is fixed by $σ .$
(b)	For each $w \in W_{σ},$ there exists $n_{w} \in N_{σ},$ indeed $n_{w} \in ⟨ U_{σ}, U_{σ}^{-} ⟩,$ so that $n_{w} H = w .$
(c)	If $n_{w}$ $(w \in W_{σ})$ is as in (b), then $G_{σ} = ⋃_{w \in W_{σ}} B_{σ} n_{w} U_{w, σ}$ with uniqueness of expression on the right.

Proof.

(a) This is clear since $U$ and $w^{- 1} U^{-} w$ are fixed by $σ .$

(b) We may assume that $w = w_{π}$ for some $ρ -orbit$ of simple $π .$ By Lemma 61, choose $x \in 𝔛_{- a, σ} x \neq 1,$ where $a \in Σ / R$ corresponds to $π .$ Using Theorem 4' we may write $x = u n_{w} v$ for some $w \in W$ where $u \in U,$ $v \in U_{w},$ and $n_{w} H = w .$ Now $x = σ x = σ u \cdot σ n_{w} σ v$ and by Theorem 4 and the uniqueness in Theorem 4', we have $σ w = w,$ $σ n_{w} = n_{w},$ $σ u = u,$ and $σ v = v .$ Thus, $n_{w} \in ⟨ U_{σ}, U_{σ}^{-} ⟩ .$ Since $w \neq 1,$ $w \in W_{σ},$ and $w \in W_{π},$ we have $w α < 0$ for some $α \in π,$ $w π < 0,$ and $w = w_{π} .$

(c) Let $x \in G_{σ},$ say $x \in B w B,$ Since $σ (B w B) = B σ w B$ we have $w \in W_{σ} .$ Choose $n_{w}$ as in (b) and write $x = b n_{w} v$ with $b \in B$ and $v \in U_{w} .$ Applying $σ$ we get $b \in B_{σ}$ and $v \in U_{w, σ} .$ Uniqueness follows from Theorem 4'.

$□$

Corollary: The conclusions of Theorem 33 are still valid if $G_{σ}$ and $B_{σ}$ are replaced by $G_{σ}^{'} = ⟨ U_{σ}, U_{σ}^{-} ⟩$ and $B_{σ}^{'} = G_{σ}^{'} \cap B_{σ} .$ Also since $B_{σ} = U_{σ} H_{σ},$ we can replace $H_{σ}$ by $H_{σ}^{'} = G_{σ}^{'} \cap H_{σ} .$

Lemma 62: Let $a$ generically denote a class in $Σ / R .$ Let $S$ be a union of classes in $Σ / R$ which is closed under addition and such that if $a \subseteq S$ then $- a ⊈ S .$ Then $𝔛_{S, σ} = \underset{α \subseteq S}{Π} 𝔛_{a, σ}$ with the product taken in any fixed order and there is uniqueness of expression on the right. In particular, $U_{σ} = \underset{a > 0}{Π} 𝔛_{α, σ}$ and $U_{w, σ} = \underset{\underset{w a \leq 0}{a > 0}}{Π} 𝔛_{a, σ}$ for all $w \in W_{σ} .$

Proof.

We arrange the positive roots in a manner consistent with the order of the $a's;$ i.e., those roots in the first $a$ are first, etc. Now $𝔛_{S} = \underset{α > 0}{Π} 𝔛_{α}$ in the order just described and with uniqueness of expression on the right by Lemma 17. Hence $𝔛_{S} = \underset{a > 0}{Π} 𝔛_{a}$ in the given order and again with uniqueness of expression on the right. The lemma follows by considering the fixed points of $σ$ on both sides of the last equation.

$□$

Corollary: If $a, b$ are classes in $Σ / R$ with $a \neq \pm b,$ then $(𝔛_{a}, 𝔛_{b}) \subseteq Π 𝔛_{c},$ where the roots on the right are in the closed subsystem generated by $a$ and $b,$ those of $a$ and $b$ excluded. The condition on $c$ can be stated alternately, in terms of $Σ_{σ},$ that $\overline{c}$ is in the interior of the (plane) convex cone generated by $\overline{a}$ and $\overline{b} .$

Remark: The exact relations in the above corollary can be quite complicated but generally resemble those in the Chevalley group whose Weyl group is $W_{σ} .$ For example, if $G$ is of type $A_{3}$ and $σ$ is of order 2, say $\begin{matrix} \end{matrix},$ $a = {β},$ $b = {α, γ},$ $c = {α + β, β + γ},$ $d = {α + β + γ},$ and if we set $x_{α} (t) = x_{β} (t)$ $(t \in k_{θ}),$ $x_{b} (u) = x_{α} (u) x_{γ} (u^{θ})$ $(u \in k),$ and similarly for $c$ and $d,$ we get $(x_{a} (t), x_{b} (u)) = x_{c} (\pm t u) x_{d} (\pm t u u^{θ}) .$ In $C_{2},$ the corresponding relation is $(x_{a} (t), x_{b} (u)) = x_{a + b} (\pm t u) x_{a + 2 b} (\pm t u^{2}) .$

If $G$ is of type $X$ and $σ$ is of order $n,$ we say $G_{σ}$ is of type ${}^{n}X .$ E.g., the group considered in the above remark is of type ${}^{2}A_{3} .$ The group of type ${}^{2}C_{2}$ is called the Suzuki group and the groups of type ${}^{2}G_{2}$ and ${}^{2}F_{4}$ are called Ree groups. We write $G \sim X$ and $G_{σ} \sim {}^{n}X .$

Lemma 63: Let $a$ be a class in $Σ / R,$ then $𝔛_{a, σ}$ has the following structure:

(a)	If $a \sim A_{1},$ then $𝔛_{a, σ} = {x_{α} (t) \| t \in k_{θ}}$
(b)	If $a \sim A_{1}^{n},$ then $𝔛_{a, σ} = {x \cdot σ x \dots \| x = x_{α} (t), α \in a, t \in k}$
(c)	If $a \sim A_{2},$ $a = {α, β, α + β},$ then $θ^{2} = 1$ and $𝔛_{a, σ} = {x_{α} (t) x_{β} (t^{θ}) x_{α + β} (u) \| t t^{θ} + u + u^{θ} = 0}$

(t, u)

denotes the given element, then

(t, u) (t', u') = (t + t', u + u' - t^{θ} t') .

(d)

a \sim C_{2},

a = {α, β, α + β, α + 2 β},

then

2 θ^{2} = 1

and

𝔛_{a, σ} = {x_{α} (1) x_{β} (t^{θ}) x_{a + 2 β} (u) x_{α + β} (t^{1 + θ} + u^{θ}) | t, u \in k} .

(t, u)

denotes the given element,

(t, u) (t', u') = (t + t', u + u' + t^{2 θ} t') .

(e)

a \sim G_{2},

a = {α, β, α + β, α + 2 β, α + 3 β, 2 α + 3 β},

then

3 θ^{2} = 1

and

𝔛_{a, θ} = {x_{α} (t) x_{β} (t^{θ}) x_{α + 3 β} (u) x_{α + β} (u^{θ} - t^{1 + θ}) x_{2 α + 3 β} (v) x_{α + 2 β} (v^{θ} - t^{1 + 2 θ}) | t, u, v \in k} .

(t, u, v)

denotes the given element then

(t, u, v) (t', u', v') = (t + t', u + u' + t' t^{3 θ} m v + v' - t' u + {t'}^{2} t^{3 θ}) .

Note that in (a) and (b), $𝔛_{a, σ}$ is a one parameter group for the fields $k_{θ}$ and $k$ respectively.

Proof.

(a) and (b) are easy and we omit their proofs. For (c), normalize the parametrization of $𝔛_{α + β}$ so that $N_{α, β} = 1 .$ Then $σ x_{α} (t) = x_{β} (t^{θ}),$ $σ x_{β} (t) = x_{α} (t^{θ}),$ and $σ x_{α + β} (u) = x_{α + β} (- u^{θ}) .$ Write $x \in 𝔛_{a, θ}$ as $x = x_{α} (t) x_{β} (v) x_{α + β} (u)$ and compare the coefficients on both sides of $x = σ x$ to get (c). The proof of (d) is similar to that of (c). For (e), first normalize the signs as in Theorem 28, and then complete the proof as in (c) and (d).

$□$

Exercise: Complete the details of the above proof.

Remark: The role of the group ${S L}_{2}$ in the untwisted case is taken by the groups ${S L}_{2} (k_{θ}),$ ${S L}_{2} (k),$ ${S U}_{3} (k, θ)$ (split form), the Suzuki group, and Ree group of type $G_{2} .$

Exercise: Determine the structure of $H_{σ}$ in the case $G$ is universal.

Lemma 64: If $G$ is universal, then $G_{σ}$ is generated by $U_{σ}$ and $U_{σ}^{-}$ except perhaps for the case $G_{σ} \sim {}^{2}G_{2}$ with $k$ infinite.

Proof.

Let $G_{σ}^{'} = ⟨ U_{σ}, U_{σ}^{-} ⟩$ and let $H_{σ}^{'} = H_{σ} \cap G_{σ}^{'} .$ By the corollary to Theorem 33, it suffices to show $H_{σ} \subseteq G_{σ}^{'};$ i.e., $(*)$ $H_{σ}^{'} = H_{σ} .$ Since $G$ is universal, $H$ is a direct product of ${𝔥_{α} | α simple}$ (see the corollary to Lemma 28). These groups are permuted by $σ$ exactly as the roots are. Hence it is enough to prove $(*)$ when there is a single orbit; i.e., when $G_{σ}$ is one of the types ${S L}_{2},$ ${}^{2}A_{2},$ ${}^{2}C_{2},$ or ${}^{2}G_{2} .$ For ${S L}_{2},$ this is clear.

(1) For $x \in U_{σ} - {1},$ write $x = u_{1} n u_{2}$ with $u_{i} \in U_{σ}^{-}, i = 1, 2$ and $n = n (x) \in N \cap G_{σ}^{'} .$ Then $H_{σ}^{'}$ is generated by ${n (x) n {(x_{0})}^{- 1} | x_{0}$ a fixed choice of $x} .$ To see this let $H_{σ}^{''}$ be the group so generated. Consider $G_{σ}^{''} = U_{σ}^{-} H_{σ}^{''} \cup U_{σ}^{-} H_{σ}^{''} n (x_{0}) U_{σ}^{-} .$ This set is closed under multiplication by $U_{σ}^{-} .$ It is also closed under right multiplication by $n {(x_{0})}^{- 1} .$ This follows from $n {(x_{0})}^{- 1} = n (x_{0}^{- 1}) = n (x_{0}^{- 1}) n {(x_{0})}^{- 1} n (x_{0})$ and $n (x_{0}) U_{σ}^{-} n {(x_{0})}^{- 1} = U_{σ} \subseteq G_{σ}^{''}$ since $x = u_{1} (n (x) n {(x_{0})}^{- 1}) n (x_{0}) u_{2}$ for $x \in U_{σ} - {1} .$ We see that $G_{σ}^{''} = G_{σ}^{'},$ whence $H_{σ}^{''} = H_{σ}^{'} .$

(2) If $α$ and $β$ are the simple roots of $A_{2},$ $C_{2},$ or $G_{2}$ labeled as in Lemma 63 (c), (d), or (e) respectively, then $H_{σ}$ is isomorphic to $k^{*}$ via the map $φ : t \to h_{α} (t) h_{β} (t^{θ}) .$

(3) Let $λ$ be the weight such that $⟨ λ, α ⟩ = 1,$ $⟨ λ, β ⟩ = 0,$ let $R$ be a representation of $ℒ^{k}$ (obtained from one of $ℒ$ by shifting the coefficients to $k)$ having $λ$ as highest weight and let $v^{+}$ be a corresponding weight vector. Let $μ$ be the lowest weight of $R$ and let $v^{-}$ be a corresponding weight vector. For $x \in U_{σ} - {1},$ write $x v^{-} = f (x) v^{+} +$ terms for lower weights. Then $f (x) \neq 0$ and $H_{σ}^{'}$ is isomorphic under $φ^{- 1}$ in (2) to the subgroup $m$ of $k^{*}$ generated by all $f (x) f {(x_{0})}^{- 1} .$ To prove (3), let $x \in U_{σ} - {1}$ and write $x = u_{1} n (x) u_{2}$ as in (1). We see $x v^{-} = n (x) v^{+} +$ terms for lower weights, so $n (x) v^{-} = f (x) v^{+}$ and $n (x) n {(x_{0})}^{- 1} v^{+} = f (x) f {(x_{0})}^{- 1} v^{+} .$ If $n (x) n {(x_{0})}^{- 1} = h_{α} (t) h_{β} (t^{θ}),$ then by the choice of $λ, f (x) f {(x_{0})}^{- 1} = t$ (see Lemma 19 (c)). (3) then follows from (1).

(4) The case $G_{σ} \sim {}^{2}A_{2} .$ Here $f (x) = - u^{θ}$ and $m = k^{*} .$ To see this, we note that the representation $R$ of (3) in this case is $R : ℒ^{k} \to {s ℓ}_{3} (k)$ and if $x = x_{α} (t) x_{β} (t^{θ}) x_{α + β} (u)$ then

x \to [\begin{matrix} 1 & t & u + t t^{θ} \\ 0 & 1 & t^{θ} \\ 0 & 0 & 1 \end{matrix}] .

Thus, $f (x) = u + t t^{θ} = - u^{θ}$ by Lemma 63 (c). Thus, $m$ is the group generated by ratios of elements $(- u^{θ})$ of $k^{*}$ whose traces are norms $(t t^{θ}) .$ Let $u \in k^{*} .$ If $u^{θ} \neq u,$ set $u_{1} = {(u - u^{θ})}^{- 1},$ and if $u^{θ} = u,$ choose $u_{1} \in k^{*}$ so that $u_{1}^{θ} = - u_{1} .$ Then $u u_{1}$ and $u_{1}$ are values of $f$ (their traces are 0 or 1), so that $u \in m$ and $m = k^{*} .$

(5) The case $G_{σ} \sim {}^{2}C_{2} .$ Here $f (x) = t^{2 + 2 θ} + u^{2 θ} + t u$ and $m = k^{*} .$ To see this, first note that since the characteristic of $k$ is 2, there is an ideal in $ℒ^{k}$ "supported" by short roots. The representation $R$ can be taken as $ℒ^{k}$ acting on this ideal, and $v^{+} = X_{α + β}$ while $v^{-} = X_{- α - β} .$ Letting $x = x_{α} (t) x_{β} (t^{θ}) x_{α + 2 β} (u) x_{α + β} (u^{θ} + t^{1 + θ})$ we can determine $f (x) .$ By taking $t = 0$ in the expression for $f (x)$ and writing $v = {(v^{θ})}^{2 θ},$ we see that $m = k^{*} .$

(6) The case $G_{σ} \sim {}^{2}G_{2} .$ Here $f (x) = t^{4 + 6 θ} - u^{1 + 3 θ} - v^{2} + t^{3 + 3 θ} u + t^{1 + 3 θ} u^{3 θ} + t v^{3 θ} - t u v .$ The group $m$ is generated by all values of $f$ for which $(t, u, v) \neq (0, 0, 0),$ and it contains ${k^{*}}^{2}$ and $- 1;$ hence $m = k^{*},$ if $k$ is finite. Here the representation $R$ can be taken to be the adjoint representation on $ℒ^{k},$ $v^{+} = X_{2 α + 3 β},$ and $v^{-} = X_{- 2 α - 3 β} .$ Letting $x$ be as in Lemma 63 (e), and working modulo the ideal in $ℒ^{k}$ "supported" by the short roots, we can compute $f (x) .$ Setting $t = u = 0,$ we see that $- v^{2} \in m,$ hence $- 1 \in m$ and ${k^{*}}^{2} \subseteq m .$ If $k$ is finite $m = k^{*}$ follows from $(*)$ $- 1 \notin {k^{*}}^{2} .$ To show $(*),$ suppose $t^{2} = - 1$ with $t \in k .$ Then $t^{2 θ} = - 1,$ so $t^{θ} = \pm t$ and $t^{θ^{2}} = t .$ Since $3 θ^{2} - 1,$ we see $t = {(t^{θ^{2}})}^{3} = t^{3} .$ But $t^{3} = t^{2} t = - t,$ so $t = 0,$ a contradiction. This proves the lemma.

$□$

Corollary: If $G$ is universal, then $G_{σ}^{'} = G_{σ}$ and $H_{σ}^{'} = H_{σ}$ except possibly for ${}^{2}G_{2}$ with $k$ infinite in which case $G_{σ} / G_{σ}^{'} = H_{σ} / H_{σ}^{'} ≅ k^{*} / m$ with $m$ as in (6) above.

Remarks:

(a)

It is not known whether

m = k^{*}

always if

G_{σ} \sim {}^{2}G_{2} .

One can make the changes in variables

v \to v + t u

and then

u \to u - t^{1 + 3 θ}

to convert the form

f

in (6) to

t^{4 + 6 θ} - u^{1 + 3 θ} - v^{2} + t^{2} u^{2} + t v^{3 θ} .

Both before and after this simplification the form satisfies the condition of homogenity:

f (t, u, v) = t^{4 + 6 θ} f (1, u / t^{1 + 3 θ}, v / t^{2 + 3 θ}) if t \neq 0 .

(b)

A corollary of (3) above, is that the forms in (5) and (6) are definite, i.e.,

f = 0

implies

t = u (= v) = 0 .

A direct proof in case

f

is as in (5) can be made as follows: Suppose

0 = f (t, u) = t^{2 + 2 θ} + u^{2 θ} + t u

with one of

t, u

nonzero. If

t = 0,

then

u = 0,

so we have

t \neq 0 .

We see

f (t, u) = t^{2 + 2 θ} f (1, u / t^{2 θ + 1})

using

2 θ^{2} = 1 .

Hence we may assume

t = 1 .

Thus,

1 + u^{2 θ} + u = 0

or (by applying

θ)

u^{θ} = 1 + u .

Hence

u^{θ^{2}} = 1 + u^{θ} = u

and

u = u^{2 θ^{2}} = u^{2} .

Thus,

u = 0

1,

a contradiction. A direct proof in case

f

is as (6) appears to be quite complicated.

(c)

The form in (5) leads to a geometric interpretation of

{}^{2}C_{2} .

Form the graph

v = t^{2 + 2 θ} + u^{2 θ} + t u

k^{3}

of the form

f (x) .

Imbed

k^{3}

P^{3} (k)

projective 3-space over

k,

by adding the plane at

\infty,

and adjoin the point at

\infty

in the direction

(0, 0, 1)

to the graph to obtain a subset

Q

P^{3} (k) .

Q

is then an ovoid in

P^{3} (k);

i.e.

(1)	No line meets $Q$ in more than two points.
(2)	The lines through any point of $Q$ not meeting $Q$ again always lie in a plane.

The group

{}^{2}C_{2}

is then realized as the group of projective transformations of

P^{3} (k)

fixing

Q .

For further details as well as a corresponding geometric interpretation of

{}^{2}G_{2}

see J. Tits, Séminaire Bourbaki, 210 (1960). For an exhaustive treatment of

{}^{2}C_{2},

especially in the finite case, see Luneberg, Springer Lecture Notes 10 (1965).

Theorem 34: Let $G$ and $σ$ be as above with $G$ universal. Excluding the cases: (a) ${}^{2}A_{2} (4),$ (b) ${}^{2}B_{2} (2),$ (c) ${}^{2}G_{2} (3),$ (d) ${}^{2}F_{4} (2),$ we have that $G_{σ}^{'}$ is simple over its center.

Sketch of proof.

Using a calculus of double cosets re $B_{σ},$ which can be developed exactly as for the Chevalley groups with $W_{σ}$ in place of $W$ and $Σ / R$ (or $Σ_{σ}$ (see Theorem 32)) in place of $Σ,$ and Theorem 33, the proof can be reduced exactly as for the Chevalley groups to the proof of: $G_{σ}^{'} = 𝒟 G_{σ}^{'} .$ If $k$ has "enough" elements, so does $H_{σ}^{'}$ by the Corollary to Lemma 64 and the action of $H_{σ}^{'}$ on $𝔛_{a, σ}$ can be used to show $𝔛_{a, σ} \subseteq 𝒟 G_{σ}^{'} .$ This takes care of nearly everything. If $k$ has "few" elements then the commutator relations within the $𝔛_{a}'s$ and among them can be used. This leads to a number of special calculations. The details are omitted.

$□$

Remark: The groups in (a) and (b) above are solvable. The group in (c) contains a normal subgroup of index 3 isomorphic to $A_{1} (8) .$ The group in (d) contains a "new" simple normal subgroup of index 2. (See J. Tits, "Algebraic and abstract simple groups," Annals of Math. 1964.)

Exercise: Center of $G_{σ}^{'} = {(Center of G)}_{σ} .$

We now are going to determine the orders of the finite Chevalley groups of twisted type. Let $k$ be a finite field of characteristic $p .$ Let a be minimal such that $θ = p^{a}$ (i.e., such that $t^{θ} = t^{p^{a}}$ for all $t \in k).$ Then $| k | = p^{2 a}$ for ${}^{2}A_{n},$ ${}^{2}D_{n},$ ${}^{2}E_{6};$ $| k | = p^{3 a}$ for ${}^{3}D_{4};$ and $| k | = p^{2 a + 1}$ for ${}^{2}C_{2},$ ${}^{2}F_{4},$ ${}^{2}G_{2} .$ We can write $σ x_{α} (t) = x_{ρ α} (ε_{α} t^{q (α)})$ where $q (α)$ is some power of $p$ less than $| k | .$ If $q$ is the geometric average of $q (α)$ over each $ρ -orbit$ then $q = p^{a}$ except when $G_{σ}$ is of type ${}^{2}C_{2},$ ${}^{2}F_{4},$ or ${}^{2}G_{2}$ in which case $q = p^{a + 1 / 2} .$

Let $V$ be the real Euclidean space generated by the roots and let $σ_{0}$ be the automorphism of $V$ permuting the rays through the roots as $ρ$ permutes the roots. Since $σ_{0}$ normalizes $W,$ we see that $σ_{0}$ acts on the space $I$ of polynomials invariant under $W .$ Since $σ_{0}$ also acts on the subspace of $I$ of homogeneous elements of a given positive degree, we may choose the basic invariants $I_{j},$ $j = 1, \dots, ℓ,$ of Theorem 27 such that $σ_{0} I_{j} = ε_{j} I_{j}$ for some $ε_{j} \in ℂ$ (here we have extended the base field $ℝ$ to $ℂ).$ As before, we let $d_{j}$ be the degree of $I_{j},$ and these are uniquely determined. Since $σ_{0}$ acts on $V,$ we also have the set ${ε_{0 j} | j = 1, \dots, ℓ}$ of eigenvalues of $σ_{0}$ on $V .$ We recall also that $N$ denotes the number of positive roots in $Σ .$

Theorem 35: Let $σ, q, N, ε_{j},$ and $d_{j}$ be as above, and assume $G$ is universal. We have

(a)	$\| G_{σ} \| = q^{N} \underset{j}{Π} (q^{d_{j}} - ε_{j}) .$
(b)	The order of the corresponding simple group is obtained by dividing $\| G_{σ} \|$ by $\| C_{σ} \|$ where $C$ is the center of $G .$

Lemma 65: Let $σ, H, U,$ etc. be as above.

(a)	$\| U_{q} \| = q^{N},$ $\| U_{w, σ} \| = q^{N (w) .}$
(b)	$\| H_{σ} \| = \underset{j}{Π} (q - ε_{0 j}) .$
(c)	$\| G_{σ} \| = q^{N} Π (q - ε_{0 j}) \underset{w \in W_{σ}}{Σ} q^{N (w)} .$

where

N (w)

is the number of positive roots in

Σ

made negative by

w .

Proof.

(a) It suffices to show that $| 𝔛_{a, σ} | = q^{| a |}$ $a \in Σ / R$ by Lemma 62. This is so by Lemma 63. (b) Let $π$ be a $ρ -orbit$ of simple roots. Since $σ h_{α} (t) = h_{ρ α} (t^{q (α)}),$ the contribution to $| H_{σ} |$ made by elements of $H_{σ}$ "supported" by $π$ is $(\underset{α \in π}{Π} q (α)) - 1 = q^{m} - 1$ if $m = | π | .$ Since the $ε_{0 j}'s$ corresponding to $π$ are the roots of the polynomial $X^{m} - 1,$ (b) follows. (c) This follows from (a), (b), and Theorem 33.

$□$

Corollary: $U_{σ}$ is a $p -Sylow$ subgroup.

Lemma 66: We have the following formal identity in $t :$

\underset{w \in W_{σ}}{Σ} t^{N (w)} = \underset{j}{Π} (1 - ε_{j} t^{d_{j}}) / (1 - ε_{0 j} t)

Proof.

We modify the proof of Theorem 26 as follows:

(a)	$σ$ there is replaced by $σ_{0}$ here.
(b)	$Σ$ there is replaced by $Σ_{0}$ here, where $Σ_{0}$ is the set of unit vectors in $V$ which lie in the same directions of the roots.
(c)	Only those subsets $π$ of $Π$ fixed by $σ_{0}$ are considered.
(d)	${(- 1)}^{π}$ is now defined to be ${(- 1)}^{k}$ where $k$ is the number of $σ_{0}$ orbits in $π .$
(e)	$W (t)$ is now defined to be $\underset{w \in W_{σ}}{Σ} t^{N (w)} .$

With these modifications the proof proceeds exactly as before through step (5). Steps (6)-(8) become:

(6')	For $π \subseteq Π,$ $w \in W,$ let $N_{π}$ be the number of cells in $K$ congruent to $D_{π}$ under $W$ and fixed by $w σ_{0} .$ Then $Σ {(- 1)}^{π} N_{π} = det w .$ (Hint: If $V' = V_{w σ_{0}}$ and $K'$ is the complex on $V'$ cut by $K,$ then the cells of $K'$ are the intersections with $V'$ of the cells of $K$ fixed by $w σ_{0} .)$
(7')	Let $x$ be a character on $⟨ W, σ_{0} ⟩$ and $x_{π}$ the restriction of $x$ to $⟨ W_{π}, σ_{0} ⟩$ induced up to $⟨ W, σ_{0} ⟩ .$ Then $Σ {(- 1)}^{π} x_{π} (w σ_{0}) = x (w σ_{0}) det w$ $(w \in W) .$
(8')	Let $M$ be a $⟨ W, σ_{0} ⟩$ module, let $\hat{I} (M)$ be the space of skew invariants under $W,$ and let $I_{π} (M)$ be the space of invariants under $W_{π} .$ Then $Σ {(- 1)}^{π} tr (σ_{0}, I_{π} (M)) = tr (σ_{0}, \hat{I} (M)) .$ The remainder of the proof proceeds as before.

$□$

Lemma 67: The $ε_{j}'s$ form a permutation of the $ε_{0 j}'s.$

Proof.

Set $t = 1$ in Lemma 66. Then $(*)$ 1 has the same multiplicity among the $ε_{j}'s$ as among the $ε_{0 j}'s.$ This is so since otherwise the right side of the expression would have either a root or a pole at $t = 1 .$ Assume $σ_{0} \neq 1,$ then either $σ_{0}^{2} = 1$ and all $ε's$ not $1$ are $- 1$ or else $σ_{0}^{3} = 1$ and all $ε's$ not 1 are cube roots of 1, coming in conjugate complex pairs since $σ_{0}$ is real. Thus in all cases $(*)$ implies the lemma.

$□$

Proof of Theorem 35.

(a) follows from Lemmas 65, 66, 67. Now let $C'$ be the center $G_{σ} .$ Clearly $C' \supseteq C_{σ} .$ Using the corollary to Theorem 33 and an argument similar to that in the proof of Corollary 1(b) to Theorem 4', we see $C' \subseteq H_{σ} \subseteq H .$ Since $H$ acts "diagonally," we have $C' \subseteq C,$ hence $C' = C_{σ},$ proving (b).

$□$

Corollary: The values of $| G_{σ} |$ and $| C_{σ} | = | Hom (L_{0} / L_{1}, k^{*}) σ |$ are as follows:

\begin{matrix} G_{σ} & ε_{j}'s \neq 1 & | G_{σ} | & | C_{σ} | \\ Chevalley group (σ = 1) & None & (*) q^{N} Π (q^{d_{k}} - 1) & | Hom (L_{0} / L_{1}, k^{*}) | \\ {}^{2}A_{n} (n \geq 2) & - 1 if d_{j} is odd & Replace q^{d_{j}} - 1 by q^{d_{j}} - {(- 1)}^{d_{j}} in (*) & Same change; i.e. (n + 1, q + 1) \\ {}^{2}E_{6} & Same as {}^{2}A_{n} & Same change as {}^{2}A_{n} & (3, q + 1) \\ {}^{2}D_{n} & - 1 for one d_{j} = n & Replace one q^{n} - 1 by q^{n} + 1 in (*) & (4, q^{n} + 1) \\ {}^{3}D_{4} & ω, ω^{2} for d_{j} = 4, 4 & q^{12} (q^{2} - 1) (q^{6} - 1) (q^{8} + q^{4} + 1) & 1 \\ {}^{2}C_{2} & - 1 for d_{j} = 4 & q^{4} (q^{2} - 1) (q^{4} + 1) & 1 \\ {}^{2}G_{2} & - 1 for d_{j} = 6 & q^{6} (q^{2} - 1) (q^{6} + 1) & 1 \\ {}^{2}F_{4} & - 1 for d_{j} = 6, 12 & q^{24} (q^{2} - 1) (q^{6} + 1) (q^{8} - 1) (q^{12} + 1) & 1 \end{matrix}

Here $ω$ denotes a primitive cube root of $1 .$

Proof (except for

| C_{σ} |).

We consider the cases:

${}^{2}A_{n} .$ We first note $(*)$ $- 1 \in W σ_{0} .$ To prove $(*)$ we use the standard coordinates ${ω_{i} | 1 \leq i \leq n + 1}$ for $A_{n} .$ Then $σ_{0}$ is given by $ω_{i} \to - ω_{n + 2 - i} .$ Since $W$ acts via all permutations of ${ω_{i}},$ we see $- 1 \in W σ_{0} .$ Alternatively, since $W$ is transitive on the simple systems (Appendix II.24), there exists $w_{0} \in W$ such that $w_{0} (- Π) = Π .$ Hence, $- w_{0} (- 1) = 1$ or $σ_{0};$ i.e., $- 1 \in W$ or $- 1 \in W σ_{0} .$ Since there are invariants of odd degree $(d_{i} = 2, 3, \dots),$ $- 1 \notin W .$ By $(*)$ $σ_{0}$ fixes the invariants of even degree and changes the signs of those of odd degree.

${}^{2}E_{6}, {}^{2}D_{2 n + 1} .$ The second argument to establish $(*)$ in the case ${}^{2}A_{n}$ may be used here, and the same conclusion holds.

${}^{2}D_{n}$ $(n$ even or odd). Relative to the standard coordinates ${v_{i} | 1 \leq i \leq n},$ the basic invariants are the first $n - 1$ elementary symmetric polynomials in ${v_{i}^{2}}$ together with $Π v_{i},$ and $W$ acts via all permutations and even number of sign changes. Here $σ_{0}$ can be taken to be the map $v_{i} \to v_{i}$ $(1 \leq i \leq n - 1),$ $v_{n} \to - v_{n} .$ Hence, only the last invariant changes sign under $σ_{0} .$

${}^{3}D_{4} .$ The degrees of the invariants are 2, 4, 6, and 4. By Lemma 67, the $ε_{j}'s$ are 1, 1, $ω,$ $ω^{2} .$ Since $σ_{0}$ is real, $ω$ and $ω^{2}$ must occur in the same dimension. Thus, we replace ${(q^{4} - 1)}^{2}$ in the usual formula by $(q^{4} - ω) (q^{4} - ω^{2}) = q^{8} + q^{4} + 1 .$

${}^{2}C_{2},$ ${}^{2}G_{2} .$ In both cases the $ε_{j}'s$ are 1, $- 1$ by Lemma 67. Since $⟨ W, σ_{0} ⟩$ is a finite group, it fixes some nonzero quadratic form, so that $ε_{j} = 1$ for $d_{j} = 2 .$

${}^{2}F_{4} .$ The degrees of the invariants are 2, 6, 8, 12 and the $ε_{j}'s$ are 1, 1, $- 1,$ $- 1 .$ As before there is a quadratic invariant fixed by $σ_{0} .$ Consider $I = \underset{α long root}{Σ} α^{8} + \underset{β short root}{Σ} {(\sqrt{2} β)}^{8} .$ We claim that $I$ is an invariant of degree 8 fixed by $σ_{0}$ and there is a quadratic invariant fixed by $σ_{0}$ which does not divide $I .$ The first part is clear since $W$ and $σ_{0}$ preserve lengths and permute the rays through the roots. To see the second part, choose coordinates ${v_{i} | i = 1, 2, 3, 4}$ so that the long roots (respectively, the short roots) are the vectors obtained from $2 v_{1},$ $v_{1} + v_{2} + v_{3} + v_{4}$ (respectively, $v_{1} + v_{2})$ by all permutations and sign changes. The quadratic invariant is $v_{1}^{2} + v_{2}^{2} + v_{3}^{2} + v_{4}^{2} .$ To show that this does not divide $I,$ consider the sum of those terms in $I$ which involve only $v_{1}$ and $v_{2}$ and note that this is not divisible by $v_{1}^{2} + v_{2}^{2} .$ Hence, I can be taken as one of the basic invariants, and $ε_{j} = 1$ if $d_{j} = 8 .$

$□$

Remark: $| {}^{2}C_{2} |$ is not divisible by 3. Aside from cyclic groups of prime order, these are the only known finite simple groups with this property.

Now we consider the automorphisms of the twisted groups. As for the untwisted groups diagonal automorphisms and field automorphisms can be defined.

Theorem 36: Let $G$ and $σ$ be as in this section and $G_{σ}^{'}$ the subgroup of $G$ (or $G_{σ})$ generated by $U_{σ}$ and $U_{σ}^{-} .$ Assume that $σ$ is not the identity. Then every automorphism of $G_{σ}^{'}$ is a product of an inner, a diagonal, and a field automorphism.

Remark: Observe that graph automorphisms are missing. Thus the twisted groups cannot themselves be twisted, at least not in the simple way we have been considering.

Sketch of proof.

As in step (1) of the proof of Theorem 30, the automorphism, call it $φ,$ may be normalized by an inner automorphism so that it fixes $U_{σ}$ and $U_{σ}^{-}$ (in the finite case by Sylow's theorem, in the infinite case by arguments from the theory of algebraic groups). Then it also fixes $H_{σ}^{'},$ and it permutes the $𝔛_{a}'s$ (a simple, $a \in Σ / R;$ henceforth we write $𝔛_{a}$ for $𝔛_{a, σ})$ and also the $𝔛_{- a}'s$ according to the same permutation, in an angle preserving manner (see step (2)) in terms of the corresponding simple system $Π_{σ}$ of $V_{σ} .$ By checking cases one sees that the permutation is necessarily the identity: if $k$ is finite, one need only compare the various $| 𝔛_{a} |'s$ with each other, while if $k$ is arbitrary further argument is necessary (one can, for example, check which $𝔛_{a}'s$ are Abelian and which are not, thus ruling out all possibilities except for ${}^{2}A_{3},$ ${}^{2}E_{6},$ and ${}^{3}D_{4},$ and then rule out these cases (the first two together) by considering the commutator relations among the $𝔛_{a}'s).$ As in step (4) of the proof of Theorem 30, we need only complete the proof of our theorem when $G_{σ}^{'}$ is one of the groups $G_{a} = ⟨ 𝔛_{a}, 𝔛_{- a} ⟩,$ in other words, when $G_{σ}^{'}$ is of one of the types $A_{1},$ ${}^{2}A_{2},$ ${}^{2}C_{2}$ or ${}^{2}G_{2}$ (with ${}^{2}C_{2} (2)$ and ${}^{2}G_{2} (3)$ excluded, but not $A_{1} (2),$ $A_{1} (3),$ or ${}^{2}A_{2} (4)),$ which we henceforth assume. The case $A_{1}$ having been treated in §10, we will treat only the other cases, in a sequence of steps. We write $x (t, u)$ or $x (t, u, v)$ for the general element of $U_{σ}$ as given in Lemma 63 and $d (s)$ for $h_{α} (s) h_{β} (s^{θ}) .$

(1) We have the equations

\begin{matrix} d (s) x (t, u) d {(s)}^{- 1} & = & x (s^{2 - θ} t, s^{1 + θ} u) & in {}^{2}A_{2} \\ = & x (s^{2 - 2 θ} t, s^{2 θ} u) & in {}^{2}C_{2} \\ d (s) x (t, u, v) d {(s)}^{- 1} & = & x (s^{2 - 3 θ} t, s^{- 1 + 3 θ} u, s v) & in {}^{2}G_{2} . \end{matrix}

This follows from the definitions and Lemma 20(c).

(2) Let $U_{1}, U_{2}$ be the subgroups of $U_{σ}$ obtained by setting $t = 0,$ then also $u = 0 .$ Then $U_{σ} \supset U_{1} \supset U_{2} = 1$ is the lower central series $U_{σ} \supset (U_{σ}, U_{σ}) \supset (U_{σ}, (U_{σ}, U_{σ})) \supset \dots$ for $U_{σ}$ if the type is ${}^{2}A_{2}$ or ${}^{2}C_{2},$ while $U_{σ} \supset U_{1} \supset U_{2} \supset 1$ is if the type is ${}^{2}G_{2} .$

Exercise: Prove this.

(3) If the case ${}^{2}A_{2} (4)$ is excluded, then $d (s) x (t, \dots) d {(s)}^{- 1} = x (g (s) t, \dots),$ with $g : k^{*} \to k^{*}$ a homomorphism whose image generates $k$ additively.

Proof.

Consider ${}^{2}A_{2} .$ By (1) we have $g (s) = s^{2 - θ},$ so that $g (s) = s$ for $s \in k_{θ} .$ Since $[k : k_{θ}] = 2,$ we need only show that $g$ takes on a value outside of $k_{θ} .$ Now if $g$ doesn't, then $s^{2 - θ} = {(s^{2 - θ})}^{θ}$ so that $s^{3} \in k_{θ},$ for all $s \in k^{*},$ whence we easily conclude (the reader is asked to supply the proof) that $k$ has at most 4 elements, a contradiction. For ${}^{2}C_{2}$ and ${}^{2}G_{2}$ the proof is similar, but easier.

$□$

(4) The automorphism $φ$ (of $G_{σ}^{'})$ can be normalized by a diagonal and a field automorphism to be the identity on $U_{σ} / U_{1} .$

Proof.

Since $φ$ fixes $U_{σ},$ it also fixes $U_{1},$ hence acts on $U_{σ} / U_{1} .$ Thus there is an additive isomorphism

f : k \to k such that φ x (t, \dots) = x (f (t), \dots) .

By multiplying $φ$ by a diagonal automorphism we may assume $f (1) = 1 .$ Since $φ$ fixes $H_{σ}^{'},$ there is an isomorphism $i : k^{*} \to k^{*}$ such that $φ d (s) = d (i (s)) .$ Combining these equations with the one in (3) we get

f (g (s) t) = g (i (s)) f (t) for all s \in k^{*}, t \in k .

Setting $t = 1,$ we get $(*)$ $f (g (s)) = g (i (s)),$ so that $f (g (s) t) = f (g (s)) f (t) .$ If the case ${}^{2}A_{2} (4)$ is excluded, then $f$ is multiplicative on $k$ by (3), hence is an automorphism. The same conclusion, however, holds in that case also since $f$ fixes 0 and 1 and permutes the two elements of $k$ not in $k_{θ} .$

$□$

Our object now is to show that once the normalization in (4) has been attained $φ$ is necessarily the identity.

(5) $φ$ fixes each element of $U_{1} / U_{2}$ and $U_{2},$ and also some $w \in G_{σ}^{'}$ which represents the nontrivial element of the Weyl group.

Proof.

The first part easily follows from (2) and (4), then the second follows as in the proof of Theorem 33(b).

$□$

(6) If the type is ${}^{2}C_{2}$ or ${}^{2}G_{2},$ then $φ$ is the identity.

Proof.

Consider the type ${}^{2}C_{2} .$ From the equation $(*)$ of (4) and the fact that $f = 1,$ we get $g (s) = g (i (s)),$ i.e., $s^{2 - 2 θ} = i {(s)}^{2 - 2 θ},$ and then taking the $1 + θ th$ power, $s = i (s);$ in other words $φ$ fixes every $d (s) .$ By (4) and (5), $φ x (t, u) = x (t, u + j (t))$ with $j$ an additive homomorphism. Conjugating this equation by $d (s) = φ d (s),$ using (1), and comparing the new equation with the old, we get $j (s^{2 - 2 θ} t) = s^{2 θ} j (t),$ and on replacing $s$ by $s^{1 + θ},$ $j (s t) s^{1 + 2 θ} j (t) .$ Choosing $s \neq 0, 1,$ which is possible because ${}^{2}C_{2} (2)$ has been excluded, and replacing $s$ by $s + 1$ and by 1 and combining the three equations, we get $(s + s^{2 θ}) j (t) = 0 .$ Now $s + s^{2 θ} \neq 0,$ since otherwise we would have $s + s^{2 θ} = {(s + s^{2 θ})}^{2 θ},$ then $s = s^{2},$ contrary to the choice of $s .$ Thus $j (t) = 0 .$ In other words $φ$ fixes every element of $U_{σ} .$ If the type is ${}^{2}G_{2}$ instead, the argument is similar, requiring one extra step. Since $G_{σ}^{'}$ is generated by $U_{σ}$ and the element $w$ of (5), $φ$ is the identity.

$□$

The preceding argument, slightly modified, barely fails for ${}^{2}A_{2},$ in fact fails just for the smallest case ${}^{2}A_{2} (4) .$ The proof to follow, however, works in all cases.

(7) If the type is ${}^{2}A_{2},$ then $φ$ is the identity.

Proof.

Choose $w$ as in (5) and, assuming $u \neq 0,$ write $w x (t, u) w^{- 1} = x n x'$ with $x, x' \in U_{σ},$ $n \in H_{σ}^{'} w .$ A simple calculation in ${S L}_{3}$ shows that $x = x (a t {\overline{u}}^{- 1}, *)$ for some $a \in k^{*}$ depending on $w$ but not on $t$ or $u .$ (Prove this.) If now we write $φ x (t, u) = x (t, u + j (t)),$ apply $φ$ to the above equation, and use (4) and (5), we get $t {\overline{u}}^{- 1} = t {(\overline{u + j (t)})}^{- 1},$ so that $j (t) = 0$ and we may complete the proof as before.

$□$

It is also possible to determine the isomorphisms among the various Chevalley groups, both twisted and untwisted. We state the results for the finite groups, omitting the proofs.

Theorem 37: (a) Among the finite simple Chevalley groups, their twisted analogues, and the alternating groups $𝒜_{n}$ $(n \geq 5),$ a complete list of isomorphisms is given as follows.

(1)	Those independent of $k .$ $\begin{matrix} C_{1} \sim B_{1} \sim A_{1} \\ C_{2} \sim B_{2} \\ D_{2} \sim A_{1} \times A_{1} & {}^{2}D_{2} \sim {}^{2}{(A_{1} \times A_{1})} \sim A_{1} \\ D_{3} \sim A_{3} & {}^{2}D_{3} \sim {}^{2}A_{3} \\ {}^{2}A_{1} (q^{2}) \sim A_{1} (q) \end{matrix}$
(2)	$B_{n} (q) \sim C_{n} (q)$ if $q$ is even.
(3)	Just six other cases, of the indicated orders. $\begin{matrix} A_{1} (4) \sim A_{1} (5) \sim 𝒜_{5} & 60 \\ A_{1} (7) \sim A_{2} (2) & 168 \\ A_{1} (9) \sim 𝒜_{6} & 360 \\ A_{3} (2) \sim 𝒜_{8} & 20160 \\ {}^{2}A_{3} (4) \sim B_{2} (3) & 25920 \end{matrix}$

(b) In addition there are the following cases in which the Chevalley group just fails to be simple.

\begin{matrix} The derived group of & B_{2} (2) \sim 𝒜_{6} & 360 \\ G_{2} (2) \sim {}^{2}A_{2} (9) & 6048 \\ {}^{2}G_{2} (3) \sim A_{1} (8) & 504 \\ {}^{2}F_{4} (2) \end{matrix}

The indices in the original group are 2, 3, 2, 2, respectively.

Remarks:

(a)	The existence of the isomorphisms in (1) and (2) is easy, and in (3) is proved, e.g., in Dieudonné (Can. J. Math. 1949). There also the first case of (b), considered in the form $B_{2} (2) \sim S_{6}$ (symmetric group) is proved.
(b)	It is natural to include the simple groups $𝒜_{n}$ in the above comparison since they are the derived groups of the Weyl groups of type $A_{n - 1}$ and the Weyl groups in a sense form the skeletons of the corresponding Chevalley groups. We would like to point out that the Weyl groups $W (E_{n})$ are also almost simple and are related to earlier groups as follows.

Proposition: We have the isomorphisms:

\begin{matrix} 𝒟 W (E_{6}) \sim B_{2} (3) \sim {}^{2}A_{3} (4) \\ 𝒟 W (E_{7}) \sim C_{3} (2) \\ 𝒟 W (E_{8}) / C \sim D_{4} (2), with C the center, of order 2. \end{matrix}

Proof.

The proof is similar to the proof of $S_{6} = W (A_{5}) \sim B_{2} (2)$ given near the beginning of §10.

$□$

Aside from the cyclic groups of prime order and the groups considered above, only 11 or 12 other finite simple groups are at present (May, 1968) known. We will discuss them briefly.

(a) The five Mathieu groups $M_{n}$ $(n = 11, 12, 22, 23, 24) .$ These were discovered by Mathieu about a hundred years ago and put on a firm footing by Witt (Hamburger Abh. 12 (1938)). They arise as highly transitive permutation groups on the indicated numbers of letters. Their orders are:

\begin{matrix} | M_{11} | = 7920 = 8 \cdot 9 \cdot 10 \cdot 11 \\ | M_{12} | = 95040 = 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \\ | M_{22} | = 443520 = 48 \cdot 20 \cdot 21 \cdot 22 \\ | M_{23} | = 10200960 = 48 \cdot 20 \cdot 21 \cdot 22 \cdot 23 \\ | M_{24} | = 244823040 = 48 \cdot 20 \cdot 21 \cdot 22 \cdot 23 \cdot 24 \end{matrix}

(b) The first Janko group $J_{1}$ discovered by Janko (J. Algebra 3 (1966)) about five years ago. It is a subgroup of $G_{2} (11)$ and can be represented as a permutation group on 266 letters. Its order is

| J_{1} | = 175560 = 11 (11 + 1) (11^{3} - 1) = 19 \cdot 20 \cdot 21 \cdot 22 = 55 \cdot 56 \cdot 57 .

The remaining groups were all uncovered last fall, more or less.

(c) The groups $J_{2}$ and $J_{2 1 / 2}$ of Janko. The existence of $J_{2}$ was put on a firm basis first by Hall and Wales using a machine, and then by Tits in terms of a "geometry." It has a subgroup of index 100 isomorphic to $𝒟 G_{2} (2) \sim {}^{2}A_{2} (9),$ and is itself of index 416 in $G_{2} (4) .$ The group $J_{2 1 / 2}$ has not yet been put on a firm basis, and it appears that it will take a great deal of work to do so (because it does not seem to have any "large" subgroups), but the evidence for its existence is overwhelming. The orders are:

\begin{matrix} | J_{2} | = 604800 \\ | J_{2 1 / 2} | = 50232960 . \end{matrix}

(d) The group $H$ of D. Higman and Sims, and the group $H'$ of G. Higman. The first group contains $M_{22}$ as a subgroup of index 100 and was constructed in terms of the automorphism group of a graph with 100 vertices whose existence depends on properties of Steiner systems. Inspired by this construction, G. Higman then constructed his own group in terms of a very special geometry invented for the occasion. The two groups have the same order, and everyone seems to feel that they are isomorphic, but no one has yet proved this. The order is:

| H | = | H' | = 44352000 .

(e) The (latest) group $S$ of Suzuki. This contains $G_{2} (4)$ as a subgroup of index 1782, and is contructed in terms of a graph whose existence depends on the imbedding $J_{2} \subset G_{2} (4) .$ It possesses an involutory automorphism whose set of fixed points is exactly $J_{2} .$ Its order is:

| S | = 448345497600 .

(f) The group $M$ of McLaughlin. This group is constructed in terms of a graph and contains ${}^{2}A_{3} (9)$ as a subgroup of index 275. Its order is:

| M | = 898128000 .

Theorem 38: Among all the finite simple groups above (i.e., all that are currently known), the only coincidences in the orders which do not come from isomorphisms are:

(a)	$B_{n} (q)$ and $C_{n} (q)$ for $n \geq 3$ and $q$ odd.
(b)	$A_{2} (4)$ and $A_{3} (2) \sim 𝒜_{8} .$
(c)	$H$ and $H'$ if they aren't isomorphic.

That the groups in (a) have the same order and are not isomorphic has been proved earlier. The orders in (b) are both equal to 20160 by Theorem 25, and the groups are not isomorphic since relative to the normalizer $B$ of a $2 -Sylow$ subgroup the first group has six double cosets and the second has 24. The proof that (a), (b) and (c) represent the only possibilities depends on an exhaustive analysis of the group orders which can not be undertaken here.

Notes and References

This is a typed excerpt of Lectures on Chevalley groups by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson. This work was partially supported by Contract ARO-D-336-8230-31-43033.

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