représentation de steinberg et identités de projecteurs

L
Xns(p)X0(p2)new X+
ns(p)X+
0(p2)new
Γ\H H
ΓGL(2,Z)
Γ0(p2) Γns(p)p GL(2,Fp)
GL(2,Fp)
Γ\H'G(Q)\G(A)/Z∞K∞K .
G GL(2) HK G(Af)
det(K) = Z×ΓG∞prG∞(G(Q)∩(G+
∞×K))
G(A)
k π =⊗vπvπ∞Dk
G∞k
ΓN KNTn
n N hn

G
trace f(k)
∞⊗eN⊗hnkL2 0(G(Q)\G(A))
Sk(Γ) kΓf(k)
∞
GL(2) Dk
π∞G∞
trace π∞(f(k)
∞) = (1π∞'Dk
0
n p
trace (Tn|S2(Γns(p))) −trace Tn|S2(Γ0(p2))new= trace (Tn|S2(Γ(1))) = 0 ,
new
Γ Γ+
GL(2,Z)
Sk(Γ0(pn))
z7→ f(pdz)f
Sk(Γ0(pm)) 0 ≤m≤n d n −m wn
T Sk(Γ0(pn)) vn
wn=
n
X
m=0
vm(n−m+ 1) Xwntn=Pvntn
(1 −t)2vn=wn−2wn−1+wn−2
wm= 0 m < 0
Γ0(p2) Γs(p)
f(2)
∞⊗fp⊗hnL2
0(G(Q)\G(A))
fp=eKns −(eKs−2eK0+eK)−eK=eKns −eKs+ 2eK0−2eK.
det(Kns =Zp
KsKns K+
s=Ks∪wsKsK+ ns =Kns ∪wnsKns
ws= 0 1
1 0 !wns = 1 0 0−1!.
fpf+
p
f+
p=eK+
ns −eK+ s+eK0−eK.
2
trace πp(fp) = trace πp(f+
p) = 0
πpGpf
GpfKKtrace πp(fK) = vol(K) trace πp(f)

f fpFpf K
fK= 0 .
p
∀K1< H < K ∀γ∈K eK
H(γ) = [G:H]card(Oγ∩H)
card(Oγ)
x G
[G:Ts] card(Ox∩Ts)−2[G:B] card(Ox∩B) + 2 card(Ox)−[G:Tns] card(Ox∩Tns) = 0
[G:Ts] card(Ox∩wsTs= [G:Tns] card(Ox∩wnsTns).
GL(2,Fp)
G k k[G]G k(G)0⊂k[G]
A G [A]⊂k[G]Pa∈Aa∈k[G]pA= [A]/|A|
A G
ϕ:k[G]→k(G)0, f 7→ X
g∈G
g−1fg.
f f0k[G]f≡f0ϕ(f) = ϕ(f0)
card G k p1p2k[G]
ϕ(p1) = ϕ(p2)
G= GL(2) FB
GL(2) T⊂B U ⊂B T 0
F
N N0NG(T)NG(T0)W W 0
N/T N0/T 0
G
[G] = 1
card NG(T)ϕ([T]) + 1 card NG(T0)ϕ([T0]) + 1 card NG(U)ϕ([ZU ]) + 1 card Gϕ([Z])
B T 0
card G k pT0+pB−2pG
pT−pBk[G]Q[G]
Q[G/T ]⊕Q⊕Q Q[G/T 0]⊕Q[G/B]⊕Q[G/B]

F
q p G F
r rsF F F
1HH B G
StGG(F)R(G)
G
StG=X
P⊃B
(−1)rs(P)IndG
P1P
G B
StG=X
T
1
|W(T)|(−1)r(G)−r(T)IndG
T1T
G
G G G =SL(2) G=U(2)
SL(2)
B
G T G B σ
Gal(F /F )F F
G(F)
g−1T g g G(F)σ(g)g−1
T
Tww W (T)Twg−1T g
σ(g)g−1=w
L P
G RG
LL G
P
G
Qlw W =W(T)Tww(12.13)
1G=1
|W|X
w∈W
RG
Tw1TwStG=1
|W|X
w∈W
RG
Tw1Tw.StG.
(12.18) (12.10)
(−1)r(G)RG
Tw1TwStG= (−1)r(Tw)IndG Tw1Tw(−1)r(G)(−1)r(Tw)= (−1)l(w),
StG=1
|W|X
w∈W
(−1)l(w)IndG
Tw1Tw.

G=GL(2)
GL(2) pT−pBpT0+pB−2pG
Q[G]
X/T X/T 0X X(p)
pThpT0hQ[G] (pT−pB)h=
h(pT0+pB−2pG)
h
pThpT0=pTpT0
π(pT)π(pT0)6= 0
π G(F)T(F)T0(F)
(T, T 0)
T
T0=T0
εT0 ε(F)
t0(a, b) = a εb
b a !
a b F ε F
N(T)(F)∩T0(F)
G(F)N(T0)(F)∩T(F)w
t0(0,1) π G(F)π(pT)π(pT0
ε)
π(w)
π T (F)N(T)(F)
(T, T 0)pT−pBpT0+pB−2pGhQ[G]
pThpT0=pTpT0
T0∩B=Z G(F) = T(F)U(F)T0
ε(F)
T0=tuT 0
εu−1t−1t T (F)u U(F)
pTpT0=pTupT0
εu−1t−1.
π(pT)π(pT0)π(pT)π(u)π(pT0
ε)T0=
tuT 0
εu−1t−1h
π G(F)T(F)T0(F)π(pTupT0
ε)6= 0
u∈U(F)
π G(F)T(F)T0
ε(F)π(pTupT0
ε)6= 0
F
u U(F)
u u2
pTpT0
ε+pTupT0 ε+pTu2pT0 ε= 3pTpUpT0 ε= 3pG.
G(F)
π(pT)π(pT0
ε)π(pT)π(u)π(pT0 ε)π(pT)π(u2)π(pT0 ε)u u2