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

(**)