PARABOLIC SUBGROUPS AND INDUCTION
9
a e ($
+
* - $
+
*) U (-$£
- (~*+
*))• Thus the Levi factor of i s L *
J KRJ * KnJ KRJ
= LK n (Lj) °.
Now l e t B. be a root of $
+
* - $
+
* and £
0
a root of
1 J KRJ ^
- J - ( - $ + *). Write jff. a s a (positive or negative) integral combination of
* KnJ 1
the a. e A. £., must have at leas t one nonzero coefficien t where £«
h a s a
zero, or e l s e £- c $
+
* and similarly B0 has at l e a s t one nonzero coefficient
1
KnJ
2
where B1 has a zero. This shows the sum B^ + B2 has both negative and
positiv e coefficient s when expresse d in terms of A, so cannot be a root of $.
This means U. and UD commute in G (indeed, the commutator relation s of [20],
lemma 32.5 hold whenever B^ and £
2
a r e linearl y independent roots). Moreover,
the U_ for a e #
+
* - $
+
* form a subgroup U * * and similarly the
a
J KnJ J ,KnJ
other s form a subgroup U~ * . Since we have see n U * * commutes with
K,KnJ J ,KnJ
U~ *, thi s shows RTT(H£) i s a direct product U * * x U~ * so we can
K,KnJ J ,KnJ K,KnJ
explicitl y write down the Levi decomposition:
(2.1)
H
K
= L
**
( u
* *
x u
*
* KnJ J ,KOJ K,KnJ
Now let H be any algebraic group and let S be an irreducible H-module.
RTT(H)
The fixed point space S i s nonzero because Ry(H) i s unipotent, but i s an
RTJ(H)
H-submodule because Ry(H) i s normal in H. Thus S = S by irreducibility. This
shows R„(H) act s triviall y on any irreducible H-module, making S into an
irreducible H/Rrj(H)-module. Conversely, if S i s an irreducible module for H/RU(H),
the action extends to H by making R^H) act trivially , showing irreducible
H-modules are in one to one correspondence with irreducible H/RU(H) modules.
In case H has a Levi decomposition thi s say s the irreducible H-modules
are the same as the irreducibles for the Levi factor of H. For H a parabolic or
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