2 MARTIN W. LIEBECK AND GARY M. SEITZ

X = Ai

A2

B2

G2

A3

B3

c3

G = E% E7

EQ

F4 G2

7 7 5 3 3

5 5 3 3

5 3 3 2

7 7 3 2

2 2 2

2 2 2 2

3 2 2 2

2 2 2

For example, N(A2^Ej) = 5, and so on. (This is the table of [Se2, Theorem 1], with

a few additional entries.) If (X, G) is not in the table, set N(X,G) = 1. And if

X is a non-abelian closed connected reductive subgroup of G, and Xf — X\...Xt, a

commuting product of simple groups X;, then we define

N(X,G) = max(N{Xi,G) : 1 i t)

In particular, if p 7 then p N(X, G) for all X, G.

Theore m 1 Let X be a closed connected reductive subgroup of G, and assume that

either p — 0 or p N(X,G). Suppose that X lies in a parabolic subgroup P = QL

of G, with unipotent radical Q and Levi subgroup L. Then all closed complements to

Q in the semidirect product QX are Q-conjugate. In particular, X is Q-conjugate to

a subgroup of L.

In Section 3 we also obtain a version of Theorem 1 when G is a classical group

of small rank (see Theorem 3.8).

The proof of Theorem 1 is based on the fact that Q has a filtration by particular

high weight modules for L. Choosing P to be minimal (subject to containing X ) , we

find the embedding of J in I modulo Q, so we can restrict each of these modules

to X . Then if V is a composition factor of such a restriction, we show that with a

few exceptions, all closed complements to V in the semidirect product VX are V-

conjugate, and the desired conclusion follows from this. The machinery for carrying

out this proof is developed in Sections 1 and 2. The proof is similar in spirit to those

of [LS2, Theorems 2.1 and 6.1], but the representation theory involved is much more

complicated, and a great deal of calculation is required.

The following result is an immediate consequence of Theorem 1 (compare [LS2,

Theorem 8.1].

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

p = 0 or p X ( X , G). Suppose that X normalizes a closed unipotent subgroup U of

G. Then all closed complements to U in the semidirect product UX are G-conjugate.