LA SÉRIE PRINCIPALE UNITAIRE DE GL2(Qp) : VECTEURS

GL2(Qp)
GL2(Qp)
GL2(Qp)
pGL2(Qp)
(ϕ, Γ) 2
R(δ)
R
p
P+Zp
(ϕ, Γ) A+
Φ+Φ−
A0
R(δ)
R(δ)Z∗
p
A1
A+
LA(Zp)⊗δD(Zp)⊗δ−1
R(δ)
W(δ1, δ2) Stan(δ1, δ2)
∆{0}
(ϕ, Γ) L{{t}}

L{{t}}
ϕ
(ϕ, Γ)
ϕFr(R)
ϕR
Fr(R)
ϕFr(R)
L{{t}} ∆{0}
∆{0}
∆{0}2
Π(∆)
U7→ ∆U
∆(s)P1
Π(∆(s))
A+
pQpQp
GQp= Gal(Qp/Qp)QpWQp⊂GQp
GQpg∈WQpdeg(g)∈Z
g(x) = xpdeg(g)x∈Fpχ:GQp→Z∗
p
F∞=Qp(µp∞)HQp= ker χ= Gal(Qp/F∞)χ
Γ = GQp/HQp= Gal(F∞/Qp)Z∗
pa7→ σa
LQpOL
c
T(L)δ:Q∗
p→L∗
c
T(L)Lc T
x∈c T(L)QpL|x|
x∈Q∗
pp−vp(x)δ∈c
T(L)w(δ)
w(δ) = log δ(u)
log uu∈Z∗
p

GL2(Qp)
Wab
QpWQpQ∗
p
c
T(L)
WQpg∈WQpδ∈c T(L)δ(g)
δ(g) = δ(p)−deg(g)δ(χ(g)).
δ δ O∗
Lδ
GQpδGQpw(δ)
δ x|x|
χ1χ
•GGL2(Qp)
•B=Q∗
pQp
0Q∗
p
•Z=a0
0a, a ∈Q∗
pG
•P=Q∗
pQp
0 1 
•N=1Qp
0 1 
•T=Q∗
p0
0Q∗
pA=Q∗
p0
0 1 
•P+⊂PZp−{0}Zp
0 1 
•A+=A∩P+Zp−{0}0
0 1 
A+= Φ+×A0Φ+=pN0
0 1 A0=Z∗
p0
0 1 n≥1An
1+2pnZp0
0 1 A0
•w0 1
1 0 pGL2(Qp)
V7→ Π(V)L
VGQp2
G=GL2(Qp)
•VΠ(V)
Π(V)B
•Vss =L(δ1)⊕L(δ2)Π(V)ss =B(δ1, δ2)ss ⊕B(δ2, δ1)ss
B(δi, δ3−i)B G δ3−i⊗δiχ−1
Bδ3−i⊗δiχ−1)a b
0d=δ3−i(a)δi(d)(d|d|)−1
V7→ Π(V)
RepLGQp∼
=ΦΓet(E)GQp
(ϕ, Γ) ΦΓet(E)∼
=ΦΓet(E†)∼ =ΦΓet(R)
Rf
0< vp(T)≤r(f)r(f)>0fE†
R E E †p

V∈RepLGQp(ϕ, Γ) D∈ΦΓet(E)D†∈ΦΓet(E†)
Drig ∈ΦΓet(R)D†
D Drig D=E⊗E†D†Drig =R⊗E†D†
V2GP1=P1(Qp)
DP1D†P1Drig P1
Π(V)Π(V)an
G
0→Π(V)∗⊗ωD→DP1→Π(V)→0,
0→(Π(V)an)∗⊗ωD→Drig P1→Π(V)an →0,
ωDχ−1det VQ∗
pG
D†P1DP1Drig P1
Π(V)ωD
V χ−1
Π(V)an
J∗(Π(V)an) =
H0(N, (Π(V)an)∗)
(ϕ, Γ) Drig
(ϕ, Γ) 1 RV
DP1
Drig
∆∈ΦΓet(R) 2 V(∆) V
GQpΠ(∆) Π(V(∆))an
0→Π(∆)∗⊗ω∆→∆P1→Π(∆) →0.
∆[0] E†∆ 0 b ∆[0] E
E⊗E†∆[0] ∆ = Drig ∆[0] =D†b ∆[0] =D
J∗(Π(∆)) ∆

GL2(Qp)
J∗(Π(∆)) Π(b ∆[0])
∆
J∗(Π(∆)) Π(∆)
(ϕ, Γ) 2 (ϕ, Γ)
2 (ϕ, Γ) 1
δ∈c T(L)R(δ) (ϕ, Γ) ϕ
Rδ(p)σa∈Γδ(a)R(δ)
x⊗δ x ∈Rϕ σa
ϕ(x⊗δ) = (δ(p)ϕ(x)) ⊗δ σa(x⊗δ) = (δ(a)σa(x)) ⊗δR(δ)
R1⊗δ ϕ Γ
δ
D(ϕ, Γ) 1 R
δ∈c T(L)D∼
=R(δ)
δ1, δ2∈c
T(L) Ext1(R(δ2),R(δ1)) L
1δ1δ−1
2x−ii≥0|x|xi
i≥1 Ext1(R(δ2),R(δ1)) 2
P1(L)
S(δ1, δ2,L) (δ1, δ2)c T(L)×
c
T(L)L∈Proj(Ext1(R(δ2),R(δ1))) P0(L) = {∞}
P1(L) Ext1(R(δ2),R(δ1)) 1 2 s= (δ1, δ2,L)
∆(s) (ϕ, Γ) R(δ2)R(δ1)
L(ϕ, Γ)
0→R(δ1)→∆(s)→R(δ2)→0.
S∗Ss
vp(δ1(p))+vp(δ2(p)) = 0 et vp(δ1(p)) >0.
s∈S∗(L)s u(s)∈Q+w(s)∈L
u(s) = vp(δ1(p)) = −vp(δ2(p)) et w(s) = w(δ1)−w(δ2).
S∗S∗=Sng
∗`Scris
∗`Sst ∗`Sord ∗`Sncl ∗
•Sng
∗s w(s)≥1
•Scris
∗s w(s)≥1u(s)< w(s)L=∞
•Sst
∗s w(s)≥1u(s)< w(s)L6=∞
•Sord
∗s w(s)≥1u(s) = w(s)
•Sncl
∗s w(s)≥1u(s)> w(s)