Groupes algébriques affines 1 Généralités

k0ks
Rep(G)G
G G0
N G G/N
k G
G G0=G Gi+1 = (Gi)0Gn= 1 n G
G
G
RG RG = 1 G G/RG
G
Rep(G)G
G G
Ga
RuG RG RuG= 1 G
G/RuG G G/RuG
G G/RuG RuG
H G D
E
X G f ∈k[X]
g.f := f(g−1·)
k[X]α:k[X]→k[G]⊗k[X]α(f) = Pm
i=1 ai⊗bi
ai∈k[G]bi∈k[X]g.f =Pai(g−1)bi∈Vect(b1, . . . , bm)

f1, . . . , fnI⊂k[G]H G
G V ⊂k[G]W=V∩I I
H={g∈G, gW =W}V I H W
gW =W W I
gI =I g ∈H
d= dim(W)E=∧dV D =∧dW
H={g∈G, gD =D}gD =D W ∩gW
{ei}V W = Vect(e1, . . . , ed)gW = Vect(en, . . . , en+d−1)
n≥1g(e1∧ · · · ∧ ed)en∧ · · · ∧ en+d−1gD =D n = 1
gW =W g ∈H
H D H E
E E ⊕F F
M k[M]M
k[M]PammPamXm
I⊂M (m)=1
∆(m) = m⊗m i(m) = m−1
k A D(M)(A) = HomGp(M, A×)
Spec(k[M]) D(M)
M D(M)Gm
µn
ϕ:M→A×
k[M]→APamm7→ Pamϕ(m)
D(M)M
D(Z) = GmD(Z/nZ) = µn
D(M)
G=D(M)X(G) = Hom(G, Gm)
M D X
k[z, z−1]→k[M]
zPammPam= 1
Pamm⊗m=Pamanm⊗n
am1
G V
V=⊕χ∈X(G)VχVχV G χ
G G ⊗ks
G⊗ksGm,ksGGm,k
Γks/k G 7→ X(G⊗ks)
M7→ Dks(M)/Γ = Spec(ks[M]Γ)

Γ
k=RM=ZΓ = Z/2Z
G=GmR
C[z, z−1]Z/2Z=P∈C[z, z−1], P (z−1) = P(z).
x=1
2(z+z−1)y=1 2(iz −iz−1)
x2+y2= 1 G= Spec(R[x, y]/(x2+y2−1))
K⊂ksΓK= Gal(ks/K)
G(K) = Hom(Spec(K), GK) = Hom(Spec(ks), G ⊗ks)ΓK= HomGp(M, (ks)×)ΓK.
G(R) = Hom(Z,C×)Z/2Z={z∈C×, z−1=z}=S1
G=D(M)
M
GmGa
g g0=g gi+1 = [gi,gi]
g gn= 0 n
gRad g g Rad g= 0
ad : g→Der gxad(x)=[x, −]
g
κ(x, y) = tr(ad xad y)
g1
g1κ
g
Rep(g)

GgG
g
6= 1 6= 0
GRep(G)
W⊂V G G
g
G
G
SLnPSLn= SLn/µn
G G
G1, . . . , Gr
G1× · · · × Gr→G
g gi
g
giCi={g∈G, Ad(g)x=x, ∀x∈gj,∀j6=i}
gjj6=i Gi=C◦
iLie(Gi) = Lie(Ci) = gi
SLn×SLnµn
G G =G0
G
GRep(G)
G R Z◦G0
R=Z◦
G R G →Aut(R)
G R R ⊂Z◦R
G0
G/R f :G→G/R
f0:G0→G/R G0f0
G/R ker(f0)
ker(f0) = G0∩R=G0∩Z◦Z◦
ρ:G→GL(V)G V =⊕Vχ

Z◦G Z◦GL(V)
Z◦QGL(Vχ)G0⊂QSL(Vχ)Z◦
VχSL(Vχ)
G=Z◦G0Z◦=R G0G/R V
G V =⊕VχZ◦
Z◦G0VχG0G0Rep(G0)
VχG0
G
Rep(G)ρ:G→GL(V)
N ρ ρ|N
N
N G ρ :G→GL(V)
G ρ|N:N→GL(V)N
X Y N
X N g ∈G(k)gY N
XPg∈GgY G X
X X N
V N ViN
1N
V N = 1 V G
SL21 SL2
SL21G6= 1,SL2
dim(G)=1 G⊗kGaGmGm
Gdim(G)=2 R0 1
G/R 2 1
≤2
G=Gmg=gl1Rep(G)
Rep(g)g
2x.a
b=0 xa
a= 0
GRep(G)
GRep(G)
GRep(G) 1
GRep(G) 1
GRep(G) 1