Structure des points rationnels des groupes réductifs

G(k)G
k
k

k k
l/k
X= Spec(A)k
k l
Xl= Spec(A⊗kl)
X= Spec(A)l l A
k A k AkA=
Ak⊗kl Xk= Spec(Ak)k X X
k
Z X
J=J(Z)⊂A Jk=J∩AkZ
k J =√JkA Z k J =JkA
V l k Z
k V
Z k
k[Z]⊗kl
k(Z)⊗kl
k k
k
l/k Γ =
Gal(l/k)
R k
Γl R ⊗kl
γ·(f⊗λ) = f⊗γ(λ)∀γ∈Γ∀f∈R∀λ∈l
γR:R⊗kl→R⊗kl γ
Γl
B k BΓB
Γk B
V k k
A R k R =k
VΓV(R)
γ·g=γR◦g◦γ−1
A
l A ⊗kl
X k Y
X Y k Y l
YΓ
G
k

GRu(Gk)
Ru(G)
Gk
k
GRu(G)
Ru(G)
k
p k kp
kΓ = Gal(k/kp) Γ
Ru(G)k kp
Ru(G)k k
k p > 0J=J(Ru(G)kp)
k[G]⊗kkpJ
f⊗λ λ
kp−nkpn∈NRu(G)
kp−NN∈N
Gm
T X∗(T) = Homgr−alg(T, Gm)
X∗(T) = Homgr−alg (Gm, T )
X∗(T)X∗(T)Z
T k
k X∗(G)kk[G]k
k T
k X∗(G)k
k=R Gm,RRSO(2)R
R R
Gm,C
k
k
T G G
T
gGAd : S→gl(g)
g
g=M
α∈X∗(T)
gαgα={X∈g,∀t∈TAd(t)(X) = α(t)X}
G T α ∈
X∗(T)\{0}gαΦ(G, T )

T G k H
G T
H k
H k
(H∩T)◦kΦ(H, T ) Γ

G
P G
G/P P
G
B G/B
Φ(G, T )
T G
k
k
G k
G(k)
G(k)G
V k k(V)
k
V k k
V k V (k)V
k G k
k k
k
G
k
G
k
G(k)G