106:Groupe linéaire d`un espace vectoriel de

n k
E
GLn(k)
GL(E)k∗
GL(E)
GL(E)'SL(E)ok∗
Gln(k)GLm(k)n=m
GL(E)
Id, −Id
P GL(E) := GL(E)/Z(GL(E)) GL(E)/(λId)
P SL(E) := SL(E)/Z(SL(E)) = SL(E)/(λId)λ
n−imes k∗
SL(E)GL(E)
E f(x) = 0 u
H H
det(u)=1
Im(u−iD)⊂H
E/H
a H u(x) = x+f(x)a
E f(x) = 0 u
H H
1
Im(u−Id)6⊂ H

λ
x y u
v u(x) = y uv(x) = y
SL(E)GL(E)
SL(E)dim(E)>3n
n+ 1
P SL(n, k)k n >3n= 2
k F2
D(GL(n, k)) = D(SL(n, k)) = SL(n, k)n>3n= 2 k
F2F3
n P SL SL
GLn(R)
GLn(R)
|GLn(Fq)|= (qn−1) . . . (qn−qn−1)
|SLn(Fq)|= (qn−1) . . . (qn−qn−2)qn−1=: N
|GLn(Fq)|=N
|GLn(Fq)|=N/pgcd(N, q −1)
GL(2,F2) = SL(2,F2) = P SL(2,F2)' S3.
P GL(2,F3)' S4;P SL(2,F3)' A4.
P GL(2,F3) = P GL(2,F3)' A5.
P GL(2,F5)' S5;P SL(2,F5)' A5.

GLp(A)×GLq(A)Mp,q(A)ϕ(P, Q, M)7→
P MQ−1
∃(P, Q)B=P AQ−1
E F B0B0
0
E B1B0
1F P Q
A f B0B1
f Q−1AP
r Ir
r0
GLp(A)Mp(A)ϕ: (P, M)7→ P M P −1
∃P B =P AP −1
2 3
k2
q
q O(q)u q(u(x) = q(x)O+(q)
det(u) = 1
F g gug−1
g(F)
O(q)
n>3So(q)

(P, q)q(e1)=0, q(e2)=0 f(e1, e2)6= 0
(P, q)
q
E
GLn(R)S++
n×On(R)
2
O(E)
cos(θ)−sin(θ)//sin(θ)cos(θ)θ
1 0//0−1
O(E)SO(E)
θ θ0θ+θ0
SO(E) 2kπ/n τ r
τ0u=ττ 0τuτ −1=u−1
O−(E) 2
Gln(R)
Rn
K
GLn(R)On(R)
GLn(R)
N E
N
G O(E)
G⊂SO(E)G2kπ/n G
n Z/nZ
G6⊂ So(E)G Z/2Z
Id
O(E)

G X G
X ω X |X|=P
x∈ω
|G|
Stab(x)
u U
G SO(3, R)
n X G
X2(n−1)
G X g.x =g(x)
SO(3, R)
GL(3, R)
2
Gln(C)
GLn(C)
Gln(C)
Gln(C)
(R, +) Gln(C)
etA Gln(C)
Gln(R)