2 MARTIN W. LIEBECK AND GARY M. SEITZ X = Ai A2 B2 G2 A3 B3 c 3 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.
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