PARABOLIC SUBGROUPS AND INDUCTION
13
Corollary 3.3. If V i s an irreducible Pj-module, then there e x i s t s a unique
B-stable lin e L in V with L = Vx for some X A(V) n A+. Moreover, al l
u £ A(V) - {X} are s t r i c t l y l e s s than X in the J-relativ e partial order of
proposition 3.2.
Proof. L e x i s t s vi a the Lie-Kolchin theorem, and i s T-stable so spanned by a
weight vector v £ V. L i s U-stable s o v i s fixed by U (since the only eigenvalue
of U i s 1). By Proposition 3.2 applied to V' = V, the other statements follow. For
the uniqueness of L, le t L ' = V^, be another such line. Then X' * X and
X ^ X ' by symmetry s o X = X ' and L = L'.
Next i t i s shown that for each X e A+, ther e does indeed e x i s t an
irreducible Lj-module with high weight X. Since irreducible Lj-modules are als o
the irreducible Pj-modules, i t will follow from Lemma 3.5 below that A+ i s a
parametrizing s e t for the irreducible Pj-modules.
It i s known that the reductive group Lj can be expresse d a s a central
product Lj*T where Lj i s the (semisimple) derived group of Lj and T £ T i s a
torus. This means that Lj n T i s finite, central and Lj*T i s by definition
Lj X T/{(t
#
t - 1 )|t e Lj fl T). Let Tj = T n Lj, a maximal toru s of Lj and T als o has
a central product decomposition T = Tj»T.
We wish to identify the root system * j of Lj with a subroot system of *.
Let E be the ambient Euclidean Space and le t Ej be the subspace spanned by J.
Then * j = Ej n * taking J as a base for Sj. Let e^E - Ej be the projection
extending thi s identification with Kernel Ej and e2:E Ej the orthogonal
projection to the orthogonal complement E j of Ej. Then we can write
n - zAn) + e2(?) f o r a i i n E, and e^a) = a for any a in J. Consider the
effec t of c
i
on the fundamental weights wj. If a^ t J then ui,a = 0 for al l
a e J, s o wi c Ej and s o e^wj) = 0 and z2i^±) = wi- F o r "those a^ e J, we
will concern ourselve s with the image of c^ under c
1
only. Observe that for
dj c J also, 8jj = wi,aj = e
1
(u)
i
)+c
2
(wi)/ a j = e1(wi),aj because e
2
(**)
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