Sous-groupes compacts de GL n(R)

GLn(R)
G GLn(R)
< ., . >GRnqG
G⊂ O(qG)O(qG) = {Ω∈GLn(R),∀X∈Rn,kΩXkG=kXkG} k .kG=√qG
S∈G
X, Y ∈Rn
< X, Y >S=tXSY
G GLn(R)
X, Y ∈Rn
< X, Y >G:= 1
|G|X
A∈G
< AX, AY >
Rn
< λX +Y, Z > =1
|G|X
A∈G
< A(λX +Y), AZ >=1
|G|X
A∈G
(λ < AX, AZ > +< AY, AZ >) = λ < X, Z > +< Y, Z >
< X, Y > =< Y, X >
< X, X > =1
|G|X
A∈G
< AX, AX >=1
|G|X
A∈GkAX k2≥1
|G|kXk2≥0
< X, X >= 0 kXk= 0 X= 0 k.k
G⊂ O(qG)B∈G X ∈Rn
kBX k2
G=qG(BX) =< BX, BX >G=1
|G|X
A∈G
< ABX, ABX >=1
|G|X
A∈GkABX k2=1
|G|X
C∈GkCX k2=qG(X)
G−→ G
A7−→ AB
G
VRK
V
GGL(V)∀u∈ G u(K)⊂K
K u ∈ G
NVν(x) = maxu∈G N(u(x))
G V
G V
G −→ S(V)
u7−→ σu:x7−→ u.x := u(x)

xO(x) = {u(x), x ∈ G} ν
ν(x)x
G −→ R+
u7−→ N(u(x))
G
ν
y∈ O(x)v∈ G y=v(x)ν(y) = ν(v(x)) = ν(x)
N(u(y)) = N(u◦v(x)) ≤ν(x)u◦v∈ G
N(u(x)) = N(u◦v−1(y)) ≤ν(y)u◦v−1∈ G
ν(x) = ν(y)ν ν
Gx, y ∈ V NVu
N(u(x+y)) = N(u(x) + u(y)) ≤N(u(x)) + N(u(y)) ≤ν(x) + ν(y)
{N(u(x+y)), x, y ∈ V, u ∈ G}
ν(x+y)≤ν(x) + ν(y)
λ∈Rx∈ V N
N(u(λx)) = N(λu(x)) = |λ|N(x)≤ |λ|ν(x)
ν(λx)≤ |λ|ν(x)
λ= 0
λ6= 0 µ=1
λy=λx µ
y
ν(µy)≤ |µ|ν(y)
ν1
λλx≤
1
λ
ν(λx)
|λ|ν(x)≤ν(λx)
ν(x) = 0 ∀u∈ G N(u(x)) = 0 u(x) = 0 N
id(x)=0 x= 0
ν x, y ∈ V u0∈ G
ν(x+y) = N(u0(x) + u0(y)) = N(u0(x)) + N(u0(y)) = ν(x) + ν(y)
N ν(x+y) = ν(x) + ν(y)u0(x)
u0(y)∃λ≥0u0(x) = λu0(y)x y
u0∈ G GL(V)ν
u∈ G Fu={x∈K, u(x) = x}
∩u∈G Fu6=∅
(Fu)u∈G K
u1, . . . , upG∩1≤k≤pFuk6=∅
K

K
v=1
pX
1≤k≤p
uk
a∈K v(a) = a v V
K a
NV G ν
ν(a) = ν(v(a)) = ν
1
pX
1≤k≤p
uk(a)
≤1
pX
1≤k≤p
ν(uk(a)) = 1
pX
1≤k≤p
ν(a) = ν(a)νG
ν uk(a)∃(α1, . . . , αp)≥0
Pp
k=1 αkuk(a) = v(a) = a ν(uk(a)) = ν(a)
a a a ∈ ∩1≤k≤pFuk
GK
A∈G S ∈Sn(R)ρ(A)(S) = tASA
ρ:G−→ GL(Sn)∀A, B ∈G2ρ(BA) = ρ(A)◦ρ(B)
G A.B =BA
ρ:G−→ GL(Sn)
A7−→ ρ(A) : S7−→ tASA
ρ∀A∈G∀S∈Sn(R)ρ(A)(S)∈Sn(R)
A∈G S ∈Sn(R)
tρ(A)(S) = t(tASA) = tAtSA =tASA =ρ(A)(S)
S∈Sn(R)
A∈G ρ(A)
φ(A) : Sn−→ Sn
S7−→ tA−1SA−1
A−1A∈G G GLn(R)
A, B ∈G2S∈Sn
ρ(BA)(S) = t(BA)S(BA) = tAtBSBA =tAρ(B)(S)A=ρ(A)(ρ(B)(S)) = ρ(A)◦ρ(B)(S)
ρ(BA) = ρ(A)◦ρ(B)
δ:G−→ G×G
A7−→ (A, A)
β:Mn(R)× Mn(R)−→ L(Sn(R))
(A, B)7−→ tASB
ρ=β◦δ β

{tMM, M ∈G}
G=ρ(G)GL(Sn)ρ
ρ G
G{tMM, M ∈G}
S++
n(R)K
KGA∈G M ∈G
ρ(A)( tMM) = tA(tMM)A=t(MA)(MA)∈K
K{tMM, M ∈G}K
{tMM, M ∈G}ρ(A)
ρ(A)(λS +T) = tA(λS +T)A=λtASA +tAT A =λρ(A)(S) + ρ(A)(T)
S∈K
GS∀A∈G ρ(A)(S) = tASA =S
G⊂ O(qS)S G
Ω∈G∀X∈Rn
kΩXk2
S=<ΩX, ΩX >S=t(ΩX)S(ΩX) = tXtΩSΩX=tXSX =< X, X >S=kXk2 S
< ., . >SG∈ O(qS)
v∈ L(V)u(K)⊂K K V
v K
x0∈K xkK
xk=1
k+ 1
k
X
l=0
vl(x0)
φ xφ(k)K
a∈K
v(xk) = v 1
k+ 1
k
X
l=0
vl(x0)!=1
k+ 1
k
X
l=0
(v(vl(x0)) = 1
k+ 1
k
X
l=0
vl(x0) + 1
k+ 1 vk+1(x0)−x0
=xk+1
k+ 1 vk+1(x0)−x0
v(xφ(k)) = xφ(k)+kk−→ 0
K1
k+ 1 0
v v(a) = a
VR
KV
K