CHAPTER V
Restriction an d corestrictio n o f invariant s
13. Restrictio n o f invariant s
If / : H G i s a homomorphis m o f algebrai c group s (ove r fco), any ff-torso r
T\ (ove r a n extensio n k/ko) define s a G-torso r T = T\.G. Henc e an y invarian t
a G Inv(G, H) define s a n invarian t /* a G Inv(i7, H), wher e H is an y functo r a s i n
§1; on e ha s f*a(Ti) = a(T). Whe n H i s a subgrou p o f G an d / i s the injectio n o f
H i n G , th e ma p
/*:Inv(G,H)^Inv(ff,H)
is calle d restriction (an d i s denote d b y Res
G
o r simpl y b y Re s whe n ther e i s n o
ambiguity regardin g H an d G) .
PROPOSITION
13.1 . Let f', f : H G be two homomorphisms. Assume they
are conjugate by an element x in G(ko) (i.e. , if we write i
x
fo r th e inne r automor -
phism define d b y x, w e have / ' = i
x
o / ). Then / * is equal to (/')* .
PROOF. Le t a G Inv(G , H), and le t b = f*a an d b' (f)*a b e the correspond -
ing element s o f Inv(iJ , H). W e hav e t o sho w that , i f T\ i s an y iJ-torso r ove r a n
extension k o f &o , one ha s b(T\) b'(T\). Le t T an d T' b e th e G-torsor s deduce d
from T\ vi a / , f. Th e torso r T i s the quotien t o f T\ x G by H actin g b y
h-(t,g) = (th-
1,f(h)g),
while th e actio n o f G on the righ t come s fro m
(t,0)V = (W) .
The sam e i s true fo r T" , with / replace d b y /' . On e then define s a map 7 \ x G
Ti x G b y
This define s (b y passag e t o th e quotients ) a n isomorphis m o f T ont o T" . Thi s
implies tha t a{T) a(T'). W e then hav e
b(T1) = a(T) = a(r) = b ,(T1).
13.2. Suppos e H i s a subgrou p o f G an d writ e N fo r it s normalize r i n G .
Every elemen t o f N(ko) define s a n automorphis m o f H, henc e a n automorphis m o f
Inv(#,H).
PROPOSITION.
(1) The action of iV(fc0) on Inv(if, H) factors through the quo-
tient N(k
0
)/H(k0).
(2) If a G Inv(i7, H) z s f/i e restriction of an element o/Inv(G , H)
;
then a is
fixed by N(k 0)/H(k0).
PROOF.
Thi s follow s fro m Prop . 13.1.
33
http://dx.doi.org/10.1090/ulect/028/07
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