PARABOLIC SUBGROUPS AND INDUCTION

9

a e ($

+

* - $

+

*) U (-$£

- (~*+

*))• Thus the Levi factor of H£ 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