EXCEPTIONAL ALGEBRAIC GROUPS

3

To deduce this from Theorem 1, observe that by [BT, 3.12], UX lies in a parabolic

subgroup QL of G with U Q; then any two closed complements to U in UX are

also closed complements to Q in QX, so Theorem 1 ensures that these complements

are Q-conjugate.

Taken together with [Se2, Theorem 1], Theorem 1 leads to a description of all

closed semisimple subgroups of G (see Theorems 5 and 7 below). The next result is

also a consequence of Theorem 1.

Theorem 2 Let X be a closed connected reductive subgroup of G, and assume that

p = 0orp N(X,G). Then

(i)

CG(X)°

is reductive;

(ii) if p 0 then

OP(CG(X))

— 1 (where

OP(CG(X))

denotes the largest normal

p-subgroup of

CG(X));

(Hi) if X is semisimple, then the rank of CG(X) is equal to the maximal co-rank

among subsystem subgroups of G containing X.

We also establish a result on centralizers of non-connected reductive subgroups

(see Corollary 4.5).

The determination of all closed simple subgroups of G leads to the next result.

Theorem 3 Let X be a simple closed connected subgroup of G with rank(X) 2,

and assume that either p = 0 or p is a good prime for G and p N(Xy G). Then

CL(G)(X) = L(CG(X)).

Remarks 1. The conclusion of Theorem 3 has been shown to hold also for X = A\

in [LT].

2. The analogues of Theorems 2 and 3 for classical groups are not in general true.

For example, if G — SL(V) and X is a subgroup of G such that V is indecomposable

for X with composition series 0 V\ V2 V, where V\ = V/V2, then CG(X) is

not reductive. And if G — SLn with p = n, then CL^G){G) /

L(CG(G)).

The next theorem and its corollary concern the connection between (Aut G)-

conjugacy and linear equivalence on L(G) for subgroups of G.

Theorem 4 Let X\ and X2 be closed connected simple subgroups of G of the same

type, and assume that p — 0 or p N(Xi,G). Suppose that X\ and X2 have

the same composition factors on L(G) (counting multiplicities). Then either X\ is

conjugate to X2 in Aut G, or G — E8 and Xi = X2 = A2, with both X\ and X2

lying in subsystem groups D4D4 and projecting irreducibly in each factor.