Ej. But t h i s shows CWWJ) - u*, the corresponding fundamental dominant
weight of Aj. This shows that G
has image Aj and carrie s A+ to (Aj)+/
sinc e i t take s u^ to w^ or 0. (It als o preserve s the J-relativ e partial order of
Proposition 3.2.) It als o shows that e- carrie s the rea l span of {w.* I a* e J}
isometrically onto Ej. In fact, z1 i s merely the restrictio n map X(T) X(Tj)
induced by the inclusion Tj c T. This provides another way to compute c1(X).
For example, if a^ t J then ui,a = 0 for al l a e J so w^ e X(Pj). Thus
e1(u)^) = w^|x = w i l £ IT = 0 D e c a u s e i - t i s "tne restrictio n of a character of
the semisimple group Lj. Similarly e
agree s with the restrictio n map
X(T) - X(T).
Note that c1(A) = Aj i s a lattic e in Ej, but i s not in general equal to
A D Ej. For example, if $ i s type A2 and J = {a^), then e^wj) i s not in A.
However, 2e1(u)1) = a1 i s in A reflectin g the fact that the root lattic e Qj i s
equal to Q n Ej. Similarly, e2(A) i s a lattic e in Ej containing each o
aj t J, i.e. e2(A)
X(Pj). Again the two are not necessaril y the same. For
type A2 with J {a1 then z2(o1) = "i - e1(u1) = (l/2)w
t X(Pj).
If X = S = niiji± i s a n v w e i 9 h t o f A, write X = e^X) + G2(X) where
c1(X) = 2
n±^± a n d
G2(X) = S n
- w^ + 2 n ^ . Recalling that
e l( A +) = ^AJ^+ w e c a n n o w P r o v e :
Lemma 3.4. For any X e A+, there e x i s t s an irreducible Lj-module with high
weight X.
Proof. Write X = e1(X) + e2(X) where G1(X) (Aj)+. e2(X) can be regarded a s a
one dimensional T-module where Lj = Lj»T i s the above described central
product decomposition. Because Lj i s a semisimple group with root system * j ,
there e x i s t s an irreducible Lj-module S(c1(X)) with high weight e1(X). Consider
the irreducible Lj x T-module S(c1(X)) $ e2(X). If we can show
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