Compactification de Chabauty de l`espace des sous

SLn(R)
G
G
G
G
G1G= SL3(R) SL4(R)
GSL3(R)
G
X
GS(G)G
XS(G)
X
GS(G)
X
X X
X
X
X
X X

G
XCartan(G)G
S(G) Cartan(G)SCartan(G)
G NG(A)
A G
G/NG(A)
G/H H
H=A H
H
G/A
GRG/NG(A)X
G/A
ASL3(R)/SL3(Z)A
G/A
ASL3(R)/SL3(Z)
SL3(Z) SL3(R)/A
SLn(C)
G= SLn(C)G/NG(A)
nPn−1(C)
nPn−1
GCartan(G)S
G
G
Cartan(G)S
A(G)S(G)
G G
G
(G1, . . . , Gn)G→Qn
i=1 Gi
G

SL3(R) SL4(R) Cartan(G)S=A(G)
n>7 Cartan(SLn(R))SA(SLn(R))
GCartan(G)S
G1∂∞X X
∂∞X(2) ∂∞XCartan(G)S
G ∂∞X(2)
G= SL3(R) Cartan(G)
48
Cartan(SL3(R))S
SL3(R)
SL4(R)
G
S(G)G
G
S(G)
(Hn)n∈NG H
h∈H(hn)n∈Nh
n∈Nhn∈Hn

P⊂N(hn)n∈Ph
hn∈Hnn∈P h ∈H
G A
RG G
g
Rg g
g
gGR
Cartan(G)G
S(G)G
GCartan(G)
Cartan(G) Cartan(G)SCartan(G)
S(G)
A G gGa
A r = dim aGCartan(g)
gGrr(g)
rgGGrr(g) Cartan(g)
G
GCartan(G) Cartan(g)
Cartan(g) Cartan(g)SCartan(g)
Grr(g)
η: Cartan(G)S→Cartan(g)S
H7→ Lie(H)
G
Sp(`)`g
GE={X∈g: Sp(ad X)⊂R}
{g∈G: Sp(Ad g)⊂R}η H ∈Cartan(G)S
(exp |E)−1(H)
Cartan(G)
G/NG(A)
φ:G/NG(A)→ S(G)ϕ:G/NG(A)→ Grr(g)
gNG(A)7→ gAg−1gNG(A)7→ Ad g(a)
G
φ=η−1◦ϕ ϕ

G NG(A)G NG(A)
GL(g)N0GL(g)
aGL(g)NG(A)
GL(g)/N0Grr(g)gN0g·a
ϕ G/NG(A)
X G Plats(X)
X
X P0X A
Plats(X)→ S(G)
P=g·P07→ gAg−1G P ,
GPlats(X) Cartan(G)
X
GRG
P P
gC=g⊗CgCartan(g)S
Grr(gC)rgC
gCCartan(gC)Zar
Cartan(g)S
GRCartan(gC)Zar
gC
Cartan(gC)Zar
aC=a⊗C
(Ad G)CAd G⊂GL(gC)
l∈Cartan(gC)Zar lC
Cartan(g)SlRSp(ad l(R)) ⊂R
Cartan(g)S
Cartan(g)Grr(g)
Cartan(g)S
Cartan(g)S
Grr(g)
g=sl2(R)
Cartan(g)gP(g)