X = Ai
G = E% E7
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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