autour des algèbres de tits

G
AλG

G→GL1(Aλ)
H1(k, G)
k2
≤2
K0
G

F
2
F
A F
A
F K
G
F G
G
G Fs
F ρs:Gs→GL(Vs)
F
ρs
Gal(Fs/F )
F ρ :G→GL1(C)C
F ρs
C⊗FFs'Fs(Vs)
ρ Gs→GL1(C⊗FFs)'GL(Vs)
ρsC
G
A F
Fs
Fs(Vs)FsVs
A G = SL1(A)Fs
SL(Vs) SL(Vs)→SL(∧kVs) 1 ≤k < deg A

SL1(A)→SL1(λkA)F
λkAdeg A
k
A
λkA
A⊗kA A =F(V)λkA
F(∧kV)
A K
K
F K/F σ
F Fs(A, σ)
Fs(Vs)×Fs(Vs)op
(x, yop)7→ (y, xop)Fs(Vs)op
Fs(Vs) SU(A, σ)
FsSL(Vs)Adeg A= dimFs(Vs) = 2n
SL(Vs)→SL(∧nVs)
SU(A, σ)→SL1(D(A, σ))
F(A, σ)
D(A, σ)
F K
λnA2n
nD(A, σ)⊗FK'λnA
(V, q)F
V
v⊗v−q(v)v∈V
(V, q)C(V, q)FZ/2
C0(V, q)
V
C0(V, q)F
Spin(V, q)→GL1(C0(V, q))
Spin(V, q)
V
qC(V, q)C+× C−
F

Spin(V, q)→GL1(C0(V, q)) →GL1(C±)
C0(V, q)C+C−
Spin(V, q)
(A, σ)F
A
A(A, σ)
C(A, σ)
σC(A, σ)F×F
C(A, σ) = C+× C−
Spin(A, σ)→GL1(C(A, σ)) →GL1(C±)
F
G