Algèbres de Hecke Table des matières 1 Groupes symétriques

GLn(Fp)
n≥1n
Br+
nBr B
Bn
n−1σi1≤i≤n−1
σiσj=σjσi|i−j|>1 ;
σiσjσi=σjσiσj|i−j|= 1.
n Brn
n−1
Br+
n→Brn
BrnΣn
si:= (i i + 1) 1 ≤i<n Σn

Σnsi
s2
i= 1
si
Σn
Brn//
Brn+1
Σn//Σn+1
A t A
n > 0A AnBr+
n
σi1i−1⊕M⊕1n−i−1
M=1−t t
1 0 .
M σi
M2= (1 −t)M+t.
t A M
Brn
A=Z[t, t−1]t= 1
BrnΣn
AnΣn
q z A
q
Hn(A;q, z)
A
Ti1≤i<n
T2
i=zTi+q
A[Br+
n]
q A Ti
Hn(A;q, z)A[Brn]

z=q−1
M
Hn(A;t, 1−t)
Hn(A;t, t −1) Ti−Ti
Hn(A; 1,0) A[Σn]
A[Brn]A[Br+
n]n≥2Brn
n≥3
K
n!
Hn(K;q)
q
GLn(Fp)
p
G:= GLn(Fp)W:= Σn
W G
G B G A
AEndA[G](A[B\G]) A
(t(X)) X
B\G/B
t(X)t(Y) = X
Z⊂X.Y
µ(X, Y, Z)t(Z)
X.Y
µ(X, Y, Z) = |Y∩X−1z|
|B|
z Z z
Z⊂X.Y

HomA[G](A[B\G],−)A[G]
B G A[B\G]B'A[B\G/B]
A
A[B\G/B]' −→ EndA[G](A[B\G])
X∈B\G/B
[Bg]7→ X
b∈B/(B∩x−1Bx)
[Bxbg]
x X B ∩x−1Bx
Bxb =Bxb0bb0−1∈x−1Bx b
B B ∩x−1Bx
t(X)t(Y) [B]
X
b1∈B/(B∩x−1Bx)
b2∈B/(B∩y−1By)
[Bxb1yb2].
[Bt]t
BxByB =XY
[Bz] (b1, b2)Bxb1yb2=Bz
b1yb2∈x−1Bz |B|µ(X, Y, Z)
(b1, b2)b1yb2∈X−1z
Y∩X−1z
X−1z(b1, b2)
Y=ByB
EndA[G](A[B\G])
M7→
MBA[G]
M7→ MB
G=Gn=GLn(Fp)B=Bn
W=WnG
W' −→ B\G/B
G
B W
BsiB1≤i < n
BwB BsiBwB =BsiwB l(siw)≥l(w)l
W
{s1, . . . , sn−1}BsiBwB =BsiwB ∪BwB
A
EndA[G](A[B\G]) A A

{aw}w∈Wasiaw=asiwl(siw)> l(w)
asiaw=pasiw+ (p−1)aw
AHn(A;p)EndA[Gn](A[Bn\Gn]) Tiasi
EndA[Gn](A[Bn\Gn])
A=Z(p)Zp
p
EndFp[Gn](Fp[Bn\Gn]) ' Hn(Fp; 0)
e(i)Ti
asi1≤i < n
ˆe(i) = 1 −e(i)Hn=Hn(Fp; 0) ˆe(i) = p+ 1 −e(i)
Hn(Z(p);p)
e(i)x7→ ˆx
Hn
Z(p)
enHn
e(i)en=en=ene(i)iHnM
enM=
n−1
\
i=1
e(i)M.
w0:= (i7→ n+ 1 −i)
BnLU
n(n−1)/2
w0= (1 2 . . . n)(1 2 . . . n −1) . . . (1 2 3)(1 2)
= (s1s2. . . sn−1)(s1. . . sn−2). . . (s1s2)s1
en=−aw0
l(siw0)< l(w0)i
e(i)en=en.
ene(1) = en
en= (e(1)e(2) . . . e(n−1))((e(1)e(2) . . . e(n−2)) . . . (e(1)e(2))e(1) ;
en∈ Hne(1) ene(1) = ene(1)
ene(i) = eni w0
sil(w0si)<
l(w0)