4 MARTIN W. LIEBECK AND GARY M. SEITZ
Corollary Let Xi,X 2 be closed connected simple subgroups of G, and assume that
p = 0 or p N(Xi,G). If Xi and X2 are conjugate in GL(L(G)), then either they
are also conjugate in Aut G, or G = Eg and X\ = X2 = A 2 lying in subsystem
groups D4D4.
Remark s 1. The cases with G = Eg and X\ = X2 A2 lying in D4D4 really are
exceptions to the conclusion of Theorem 4 and the corollary: Eg has two conjugacy
classes of such subgroups A2 which have the same composition factors on L(Eg).
2. Again, a result like Theorem 4 is not in general true for subgroups of classical
groups. For example, let p 0, q =
pa
1 and let X\ = X2 = SL(W), where W is a
vector space of dimension m 3 over the algebraically closed field K of characteristic
p. Embed Xx and X2 in D = SLm2(K) via the modules W g W^ and W* ® W^
respectively, (where W^9' is a Frobenius g-power twist of W). Then X\ and X2 are
not Aut D-conjugate, but have the same composition factors on L(D).
We come now to the theorems which describe the embeddings of arbitrary semisim-
ple closed connected subgroups of G. Following [LS2], we define a subsystem sub-
group of G to be a semisimple subgroup which is normalized by a maximal torus of
G. (Except for a few cases where (G,p) = (i*4,2) or (G23), subsystem subgroups
correspond to closed subsystems of the root system of G; in the exceptions, subsys-
tems of the dual of a closed subsystem are also allowed. The closed subsystems are
listed in [Ca].) In order to state the results, we need to make one further definition,
Definition Let Y Y\ . . . Yk be a commuting product of simple algebraic groups
YJ, and let X be a closed semisimple subgroup of Y. For each i, let Y{ be the simply
connected cover of Y{. If A is a subgroup of Y, write A for the group AZ(Y)/Z(Y),
and for i = 1 , . . . , k let 7r?- : X » Y{ be the ith projection map. We call the connected
preimage of Xiti in Y{ the projection of X in Y{. We say that X is essentially embedded
in Y if the following hold for all i:
(i) if Yi is of classical type, with natural module V{ (taken to be the natural
2n-dimensional symplectic module if (Y{,p) (J5n,2)), then either the projection
of X in Yi lifts to a subgroup of Yi which is irreducible on V{, or Y{ Dn and the
projection of X in Y{ lies in a natural subgroup J5
r
5
n
_
r
_ i for some r 0, irreducible
in each factor with inequivalent representations;
(ii) if Y{ is of exceptional type, then the projection of X in Yi is either Yi or
a maximal connected subgroup of Yi not containing a maximal torus (and hence is
given by [Se2, Theorem 1]).
Note that in (i) of the above definition, if Yi is of type D4 then we allow Vi to be
any of the three irreducible 8-dimensional modules with high weights Ai, A3, A4.
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