From pro-$p$ Iwahori-Hecke modules to $(\varphi,\Gamma)$-modules, II

0.41 MB·

by  Elmar Grosse-Klönne

From pro-p Iwahori-Hecke modules to (ϕ,Γ)-modules, II
by Elmar Grosse-Kl¨onne
Abstract
Let o be the ring of integers in a finite extension field of Q , let k be its residue
p
7 field. Let G be a split reductive group over Q , let H(G,I ) be its pro-p-Iwahori
1 p 0
0 Hecke o-algebra. In [2] we introduced a general principle how to assign to a certain
2
additionally chosen datum (C(•),φ,τ) an exact functor M 7→ D(Θ V ) from finite
∗ M
n
a length H(G,I0)-modules to (ϕr,Γ)-modules. In the present paper we concretely
J work out such data (C(•),φ,τ) for the classical matrix groups. We show that the

3
corresponding functor identifies the set of (standard) supersingular H(G,I0)⊗o k-

T] modules with the set of (ϕr,Γ)-modules satisfying a certain symmetry condition.
N
.
h
Contents
t
a
m
[ 1 Introduction 1

1
v 2 (ϕr,Γ)-modules 4
5
5
6 3 Semiinfinite chamber galleries and the functor D 10
0
3.1 Power multiplicative elements in the extended affine Weyl group . . . . . . 10
0
.
1 3.2 The functor D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0
3.3 Supersingular modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7
1
:
v 4 Classical matrix groups 16
i
X 4.1 Affine root system C˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
d
ar 4.2 Affine root system B˜d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Affine root system D˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
d
˜
4.4 Affine root system A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
d
˜ ˜
5 Exceptional groups of type E and E 37
6 7
5.1 Affine root system E˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6
˜
5.2 Affine root system E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7
6 Appendix 40
1 Introduction
Let o be the ring of integers in a finite extension field of Q , let k be its residue field.
p
Let G be a split reductive group over Q , let T be a maximal split torus in G, let
p
1
I be a pro-p-Iwahori subgroup fixing a chamber C in the T-stable apartment of the
0
semi simple Bruhat Tits building of G. Let H(G,I ) be the pro-p-Iwahori Hecke o-
0
algebra. Let Modfin(H(G,I )) denote the category of H(G,I )-modules of finite o-length.
0 0
From a certain additional datum (C(•),φ,τ) we constructed in [2] an exact functor M 7→
D(Θ V ) from Modfin(H(G,I )) to the category of ´etale (ϕr,Γ)-modules (with r ∈ N
∗ M 0
depending on φ). For G = GL (Q ), when precomposed with the functor of taking I -
2 p 0
invariants, this yields the functor from smooth o-torsion representations of GL (Q ) (or
2 p
at least from those generated by their I -invariants) to ´etale (ϕ,Γ)-modules which plays
0
a crucial role in Colmez’ construction of a p-adic local Langlands correspondence for
GL (Q ). In [2] we studied in detail the functor M 7→ D(Θ V ) when G = GL (Q )
2 p ∗ M d+1 p
for d ≥ 1. In [6] the situation has been analysed for G = SL (Q ). The purpose of the
d+1 p
present paper is to explain how the general construction of [2] can be installed concretely
for other classical matrix groups G (as well as for G’s of type E , E ).
6 7
Recall that C(•) = (C = C(0),C(1),C(2),...) is a minimal gallery, starting at C, in the
T-stable apartment, that φ ∈ N(T) is a ’period’ of C(•) and that τ is a homomorphism
from Z× to T, compatible with φ in a suitable sense. The above r ∈ N is just the length
p
of φ. It turns out that τ must be a minuscule fundamental coweight (at least if the
underlying root system is simple). Conversely, any minuscule fundamental coweight τ
can be included into a datum (C(•),φ,τ), in such a way that some power of τ is a power
of φ.
For G = GL (Q ) we gave explicit choices of (C(•),φ,τ) with r = 1 in [2] (there
d+1 p
are essentially just two choices, and these are ’dual’ to each other). In the present paper
we work out ’privileged’ choices (C(•),φ,τ) for the classical matrix groups, as well as for
G’s of type E , E . We mostly consider G with connected center Z. Our choices of
6 7
(C(•),φ,τ) are such that φ ∈ N(T) projects (modulo ZT , where T denotes the maximal
0 0
bounded subgroup of T) to the affine Weyl group (viewed as a subgroup of N(T)/ZT ).
0
In particular, up to modifications by elements of Z these φ can also be included into
data (C(•),φ,τ) for the other G’s with the same underlying root system, not necessarily
with connected center. We indicate these modifications along the way. Notice that the
φ ∈ N(T) considered in [2] for G = GL (Q ) does not project to the affine Weyl group,
d+1 p
only its (d+1)-st power (which is considered here) does so. But since the discussion is
essentially the same, our treatment of the case A here is very brief.
In either case we work out the behaviour of the functor M 7→ D(Θ V ) on those
∗ M
H(G,I ) = H(G,I ) ⊗ k-modules which we call ’standard supersingular’. Roughly
0 k 0 o
speaking, these are induced from characters of the pro-p-Iwahori Hecke algebra of the
corresponding simply connected group. Each irreducible supersingular H(G,I ) -module
0 k
iscontained in(andin’most’ casesisequal to)astandardsupersingular H(G,I ) -module
0 k
2
(and the very few standard supersingular H(G,I ) -modules which are not irreducible
0 k
supersingular are easily identified). We show that our functor induces a bijection between
the set of isomorphism classes (resp. certain packets of such if G = SO (Q )) of
2d+1 p
standard supersingular H(G,I ) -modules and the set of (isomorphism classes of) ´etale
0 k
(ϕr,Γ)-modules over k = k((t)) which satisfy a certain symmetry condition (depending
E
on the root system underlying G). They are direct sums of one dimensional ´etale (ϕr,Γ)-
modules, their dimension is the k-dimension of the corresponding standard supersingular
H(G,I ) -module.
0 k
The interest in ´etale (ϕr,Γ)-modules lies in their relation with Gal -representations.
Qp
For any r ∈ N there is an exact functor from the category of ´etale (ϕr,Γ)-modules to the
category of ´etale (ϕ,Γ)-modules (it multiplies the rank by the factor r), and by means of
Fontaine’s functor, the latter one is equivalent with the category of Gal -representations.
Qp
In [2] we also explained that a datum (C(•),φ) alone, i.e. without a τ as above, can be
used to define an exact functor M 7→ D(Θ V ) from Modfin(H(G,I )) to the category of
∗ M 0
´etale (ϕr,Γ )-modules, where Γ denotes the maximal pro-p-subgroup of Γ ∼= Z×. Such
0 0 p
data (C(•),φ) are not tied to co minuscule coweights and exist in abundance. We do not
discuss them here.
We hope that, beyond its immediate purposes as described above, the present paper
may also be a useful reference for explicit descriptions of the pro-p-Iwahori algebra (in
particular with respect to the various Weyl groups involved) of classical matrix groups
other than GL (we could not find such descriptions in the literature).
d
The outline is as follows. In section 2 we explain the functor from ´etale (ϕr,Γ)-
modules to ´etale (ϕ,Γ)-modules, and we introduce the ’symmetric’ ´etale (ϕr,Γ)-modules
mentioned above, foreachoftherootsystems C,B, D andA. Insection3, Lemma3.1, we
discuss the relation between the data (C(•),φ,τ) and minuscule fundamental coweights.
Our discussions of classical matrix groups G in section 4 are just concrete incarnations of
Lemma 3.1, although in neither of these cases there is a need to make formal reference
to Lemma 3.1. On the other hand, in our discussion of the cases E and E in section 5
6 7
we do invoke Lemma 3.1. We tried to synchronize our discussions of the various matrix
groups. As a consequence, arguments repeat themselves, and we do not write them out
again and again. In the appendix we record calculations relevant for the cases E and E ,
6 7
carried out with the help of the computer algebra system sage.
Acknowledgments: Iwould liketothankLaurent Berger forahelpful discussion related
to this work. I am very grateful to the referees for the careful reading of the manuscript
and the detailed suggestions for improvements.
3
2 (ϕr,Γ)-modules
We often regard elements of F× as elements of Z× by means of the Teichmu¨ller lifting. In
p p
GL (Z ) we define the subgroups
2 p
Z× 0 1+pZ 0 1 Z
Γ = p , Γ = p , N = p
0 0
0 1 ! 0 1 ! 0 1 !
and the elements
p 0 1 1 x 0 x 0
ϕ = , ν = , h(x) = , γ(x) =
0 1 ! 0 1 ! 0 x−1 ! 0 1 !
where x ∈ Z×. Let O+ = o[[N ]] denote the completed group ring of N over o. Let O
p E 0 0 E
denote the p-adic completion of the localization of O+ with respect to the complement
E
of π O+, where π ∈ o is a uniformizer. In the completed group ring k+ = k[[N ]] we
K E K E 0
put t = [ν]−1. Let k = Frac(k+) = O ⊗ k. We identify k+ = k[[t]] and k = k((t)).
E E E o E E
For definitions and notational conventions concerning ´etale ϕr-modules and ´etale (ϕr,Γ)-
modules we refer to [2].
Let r ∈ N. Let D = (D,ϕr ) be an ´etale ϕr-module over O . For 0 ≤ i ≤ r −1 let
D E
D(i) = D be a copy of D. For 1 ≤ i ≤ r −1 define ϕ : D(i) → D(i−1) to be the identity
D
e
map on D, and define ϕ : D(0) → D(r−1) to be the structure map ϕr on D. Together
D D
e
we obtain a Z -linear endomorphism ϕ on
p D
e
r−1
D = D(i).
i=0
M
e
Define an O -action on D by the formula x·((d ) ) = (ϕi (x)d ) . Then the
E i 0≤i≤r−1 OE i 0≤i≤r−1
endomorphism ϕ of D is semilinear with respect to this O -action, hence it defines on
D E
e e
D the structure of an ´etale ϕ-module over O , see [2], section 6.3.
E
e
Let Γ′ be an open subgroup of Γ, let D be an ´etale (ϕr,Γ′)-module over O . Define
E
e
an action of Γ′ on D by
γ ·((d ) ) = (γ ·d ) .
e i 0≤i≤r−1 i 0≤i≤r−1
Lemma 2.1. The Γ′-action on D commutes with ϕ and is semilinear with respect to the
D
e
O -action, hence we obtain on D the structure of an ´etale (ϕ,Γ′)-module over O . We
E E
e
thus obtain an exact functor from the category of ´etale (ϕr,Γ′)-modules to the category of
e
´etale (ϕ,Γ′)-modules over O .
E
Proof: This is immediate from the respective properties of the Γ′-action on D. (cid:3)
TheconjugationactionofΓonN allowsustodefineforeachx ∈ Z× anautomorphism
0 p
f 7→ γ(x)fγ(x−1) of k[[t]] = k+.
E
4
Lemma 2.2. For x ∈ F× and m,n ∈ Z we have
p ≥0
(1) γ(x)tnprmγ(x−1)−(xt)nprm ∈ t(n+1)prmk[[t]].
Proof: Denote again by x be the representative of x in [1,p − 1]. We compute
[ν]x −1 = x x ([ν]−1)x = x x tj, hence
j=1 j j=1 j
P (cid:0) (cid:1) P (cid:0) (cid:1)
γ(x)tγ(x−1)−xt = ([ν]x −1)−xt ∈ t2k[[t]],
γ(x)tnγ(x−1)−(xt)n ∈ tn+1k[[t]],
γ(x)tnprmγ(x−1)−(xt)nprm = (γ(x)tnγ(x−1)−(xt)n)prm ∈ t(n+1)prmk[[t]].
(cid:3)
Lemma 2.3. (a) Let D be a one-dimensional ´etale (ϕr,Γ)-module over k . There exists
E
a basis element g for D, uniquely determined integers 0 ≤ s(D) ≤ p−2 and 1 ≤ n(D) ≤
pr −1 and a uniquely determined scalar ξ(D) ∈ k× such that
ϕr g = ξ(D)tn(D)+1−prg
D
γ(x)g −xs(D)g ∈ t·k+ ·g
E
for all x ∈ Z×. Thus, one may define 0 ≤ k (D) ≤ p−1 by n(D) = r−1k (D)pi. One
p i i=0 i
has n(D) ≡ 0 modulo (p−1).
P
(b) For any given integers 0 ≤ s ≤ p − 2 and 1 ≤ n ≤ pr − 1 with n ≡ 0 modulo
(p −1) and any scalar ξ ∈ k× there is a uniquely determined (up to isomorphism) one-
dimensional ´etale (ϕr,Γ)-module D over k with s = s(D) and n = n(D) and ξ = ξ(D).
E
Proof: (a) Begin with anarbitrarybasis element g for D; thenϕr g = Fg for some
0 D 0 0
unit F ∈ k((t)) = k . After multiplying g with a suitable power of t we may assume
E 0
F = ξtm(1 + tn0F ) for some 0 ≥ m ≥ 2 − pr, some ξ ∈ k×, some n > 0, and some
0 0
F0 ∈ k[[t]] (use tprϕr = ϕrt). For g1 = (1+tn0F0)g0 we then get ϕrDg1 = ξtm(1+tn1F1)g1
for some n > n > 0 and some F ∈ k[[t]]. We put g = (1+tn1F )g . We may continue
1 0 1 2 1 1
in this way. In the limit we get, by completeness, g = g ∈ D such that ϕr g = ξtmg. It
∞ D
is clear that ξ(D) = ξ and n(D) = pr −1+m are well defined.
Next, as γ(x)p−1 (for x ∈ Z×) is topologically nilpotent (in the topological group Γ)
p
and acts by an automorphism on D, there is some unit F ∈ k[[t]] with γ(x)g = F g.
x x
∼
Thus, the action of Γ on D induces an action of Γ on k[[t]]/tk[[t]] = k, and this action
is given by a homomorphism Z× ∼= Γ → k×, x 7→ xs(D) for some 0 ≤ s(D) ≤ p − 2.
p
Computing modulo t·k+ ·g we have (with n = n(D), s = s(D), ξ = ξ(D))
E
xsg ≡ γ(x)g = ξ−1γ(x)tpr−n−1ϕr g = ξ−1(γ(x)tpr−n−1γ(x−1))γ(x)ϕr g
D D
5
(≡i) ξ−1(xt)pr−n−1γ(x)ϕr g ≡ xsξ−1(xt)pr−n−1ϕr g = xs+pr−n−1g.
D D
Here (i) follows from γ(x)tpr−n−1γ(x−1)−(xt)pr−n−1 ∈ tpr−nk[[t]], formula (1). In short,
we have g ≡ xpr−n−1g for all x ∈ F×. We deduce n ≡ 0 modulo p−1.
p
(b) Put D = k[[t]] = k+ and D∗ = Homct(k+,k). Endow D∗ with an action of k+ by
E k E E
putting (α·ℓ)(x) = ℓ(α·x) for α ∈ k+, ℓ ∈ D∗ and x ∈ D. For j ≥ 0 define ℓ ∈ D∗ by
E j
ℓ ( a ti) = a .
j i j
i≥0
X
Then {ℓ } is a k-basis of D∗, and we have tℓ = ℓ and tℓ = 0. Define ϕr : D∗ → D∗
j j≥0 j+1 j 0
to be the k-linear map with ϕr(ℓ ) = ξ−1ℓ . We then find
j prj+n
(tpr(ϕr(ℓ )))( a ti) = (ξ−1ℓ )( a ti+pr) = ξ−1a ,
j i prj+n i pr(j−1)+n
i≥0 i≥0
X X
(ϕr(tℓ ))( a ti) = ξ−1ℓ ( a ti) = ξ−1a
j i pr(j−1)+n i pr(j−1)+n
i≥0 i≥0
X X
(with a = 0 resp. ℓ = 0 for i < 0). As tprϕr = ϕrt in k+[ϕr] this means that we have
i i E
defined an action of k+[ϕr] on D∗. Next, for x ∈ F× and b,m ∈ Z we put
E p ≥0
γ(x)·ℓ−b+nPm−1pri = x−s(γ(x)tbγ(x−1))ℓ−b+nPm−1pri,
i=0 i=0
or equivalently,
(2) γ(x)·tb(ϕr)mℓ = x−s(γ(x)tbγ(x−1))(ϕr)mℓ .
0 0
We claim that this defines an action of γ on D∗. In order to check well definedness, i.e.
independence on the choice of b and m when only −b+n m−1pri is given, it is enough
i=0
to consider the equations tb(ϕr)mℓ = ξtb+nprm(ϕr)m+1ℓ and to compare the respective
0 0 P
effect of γ(x) (through formula (2)) on either side. Namely, we need to see
ξ−1x−s(γ(x)tbγ(x−1))(ϕr)mℓ = x−s(γ(x)tb+nprmγ(x−1))(ϕr)m+1ℓ ,
0 0
or equivalently,
(γ(x)tbγ(x−1))(ξ−1−(γ(x)tnprmγ(x−1))ϕr)(ϕr)mℓ = 0.
0
For this it is enough to see that ξ−1 − (γ(x)tnprmγ(x−1)ϕr) annihilates (ϕr)mℓ . Now
0
n < pr implies (n+1)prm > n m pri and hence that t(n+1)prm annihilates (ϕr)m+1ℓ =
i=0 0
ξ−m−1ℓnPmi=0pri. Therefore it iPs enough, by formula (1), to show that ξ−1 − (xt)nprmϕr
annihilates (ϕr)mℓ . But n ≡ 0 modulo (p−1) means xnprm = 1. Or claim follows since
0
ξ−1(ϕr)mℓ = tnprm(ϕr)m+1ℓ .
0 0
6
In fact, we have defined an action of k+[ϕr,Γ] on D∗. Namely, it is clear from the
E
definitions that the relations γ(x) · t = (γ(x)tγ(x−1)) · γ(x) in k+[ϕr,Γ] are respected.
E
That the actions of γ(x) andϕr onD∗ commute follows easily from the relations γ(x)ϕr =
ϕrγ(x) and tprϕr = ϕrt in k+[ϕr,Γ]. Notice that
E
γ(x)ℓ = x−sℓ for all x ∈ Z×.
0 0 p
Now passing to the dual D ∼= (D∗)∗ of D∗ yields a non degenerate (ψr,Γ)-module over
k+ with an associated ´etale (ϕr,Γ)-module D over k with s = s(D) and n = n(D) and
E E
ξ = ξ(D); this is explained in [2] Lemma 6.4. This dualization argument also proves the
(cid:3)
uniqueness of D.
Definition: Wesaythatan´etale(ϕr,Γ)-moduleDover k isC-symmetricifitadmits
E
a direct sum decomposition D = D ⊕D with one-dimensional ´etale (ϕr,Γ)-modules D ,
1 2 1
D satisfying the following conditions (1), (2C) and (3C):
2
(1) k (D ) = k (D ) for all 0 ≤ i ≤ r−1
i 1 r−1−i 2
(2C) ξ(D ) = ξ(D )
1 2
(3C) s(D )−s(D ) ≡ r−1ik (D ) modulo (p−1)
2 1 i=0 i 1
(4) k (D ) ∈/ {(0,...,0),(p−1,...,p−1)}
• 1 P
Definition: We say that an´etale (ϕr,Γ)-module D over k is B-symmetric if r is odd
E
and if D admits a direct sum decomposition D = D ⊕ D with one-dimensional ´etale
1 2
(ϕr,Γ)-modules D , D satisfying the following conditions (1), (2B) and (3B):
1 2
(1) ki(D1) = kr−1−i(D2) for all 0 ≤ i ≤ r−1, and kr−1(D1) = kr−1(D2) is even
2 2
(2B) For both D = D and D = D we have ξ(D) = r−1(k (D)!)−1 and k (D) =
1 2 i=0 i i
k (D) for all 1 ≤ i ≤ r−1
r−1−i 2 Q
(3B) s(D )−s(D ) ≡ k (D )−k (D ) modulo (p−1)
2 1 0 1 r−1 1
(4) k (D ) ∈/ {(0,...,0),(p−1,...,p−1)}
• 1
Lemma 2.4. The conjunction of the conditions (1), (2C) and (3C) (resp. (1), (2B) and
(3B)) is symmetric in D and D .
1 2
Proof: That each one of the conditions (1), (2C) and (2B) is symmetric even indi-
vidually is obvious. Now n(D) ≡ 0 modulo p−1 implies r−1k (D ) ≡ 0 modulo p−1.
i=0 i 1
Therefore s(D ) − s(D ) ≡ r−1ik (D ) and k (D ) = k (D ) for all i (condition
2 1 i=0 i 1 i 1 Pr−1−i 2
(1)) together imply s(D ) − s(D ) ≡ r−1ik (D ). Thus condition (3C) is symmetric,
1 P 2 i=0 i 2
(cid:3)
assuming condition(1). Similarly, condition(3B)issymmetric, assuming condition(1).
P
7
Definition: (i)LetS (r)denotethesetoftriples(n,s,ξ)withintegers1 ≤ n ≤ pr−2
C
and 0 ≤ s ≤ p−2 and scalars ξ ∈ k× such that n ≡ 0 modulo (p−1). Let S (r) denote
C
the quotient of S (r) bey the involution
C
r−1 r−1 r−1
e
( k pi,s,ξ) 7→ ( k pi,s+ ik ,ξ).
i r−i−1 i
i=0 i=0 i=0
X X X
(Here and in the following, in the second component we mean the representative modulo
p−1 belonging to [0,p−2].)
(ii) Let r be odd and let S (r) denote the set of pairs (n,s) with integers 1 ≤ n =
B
r−1k pi ≤ pr − 2 and 0 ≤ s ≤ p − 2 such that n ≡ 0 modulo (p − 1) and such that
i=0 i
k = k for all 1 ≤ i ≤ r−1e. Let S (r) denote the quotient of S (r) by the involution
Pi r−1−i 2 B B
r−1 r−1 e
( k pi,s) 7→ ( k pi,s+k −k ).
i r−i−1 0 r−1
i=0 i=0
X X
Lemma 2.5. (i) Sending D = D ⊕ D to (n(D ),s(D ),ξ(D )) induces a bijection
1 2 1 1 1
between the set of isomorphism classes of C-symmetric ´etale (ϕr,Γ)-modules and S (r).
C
(ii) Sending D = D ⊕ D to (n(D ),s(D )) induces a bijection between the set of
1 2 1 1
isomorphism classes of B-symmetric ´etale (ϕr,Γ)-modules and S (r).
B
Proof: This follows from Lemma 2.3. (cid:3)
Definition: Let r be even. Let S (r) denote the set of triples (n,s,ξ) with integers
D
1 ≤ n = r−1k pi ≤ pr − 2 and 0 ≤ s ≤ p − 2 and scalars ξ ∈ k× such that n ≡ 0
i=0 i
e
modulo (pP−1) and such that ki = ki+2r for all 1 ≤ i ≤ r2 −2. We consider the following
permutations ι and ι of S (r). The value of ι at ( r−1k pi,s,ξ) is
0 1 D 0 i=0 i
r2−2 e r−2 P 2r−1
(kr + kipi +kr−1p2r−1 +k0p2r + kipi +kr−1pr−1,s+ ki,ξ).
2 2
i=1 i=r+1 i=0
X X2 X
The value of ι at ( r−1k pi,s,ξ) is
1 i=0 i
rP−1 r −2 2r−1
( kr−i−1pi,s+ (kr +k0)+ (i−1)kr−i,ξ)
4 2 2
i=0 i=2
X X
if r is odd, whereas if r is even the value is
2 2
r2−2 r−2 r r r2−1
(kr−1+ kr−i−1pi+krp2r−1+kr−1p2r+ kr−i−1pi+k0pr−1,s+( −1)kr+ k0+ (i−1)kr−ipi,ξ).
2 2 4 2 4 2
i=1 i=r+1 i=2
X X2 X
8
It is straightforward to check that ι2 = id and ι ι = ι ι , and moreover that ι2 = id if r is
0 0 1 1 0 1 2
odd, but ι2 = ι if r is even. In either case, the subgroup hι ,ι i of Aut(S (r)) generated
1 0 2 0 1 D
by ι and ι is commutative and contains 4 elements. We let S (r) denote the quotient
0 1 D
of S (r) by the action of hι ,ι i. e
D 0 1
e
Definition: Let r be even. We say that an ´etale (ϕr,Γ)-module D over k is D-
E
symmetric if it admits a direct sum decomposition D = D ⊕D ⊕D ⊕D with one-
11 12 21 22
dimensional ´etale (ϕr,Γ)-modules D , D , D , D satisfying the following conditions:
11 12 21 22
(1) For all 1 ≤ i ≤ 2r −2 and all 1 ≤ s,t ≤ 2 we have ki(Dst) = k2r+i(Dst)
(2) For all 1 ≤ i ≤ r −2 we have k (D ) = k (D ) and k (D ) = k (D )
2 i 11 i 12 i 21 i 22
(3) For j = 1 and j = 2 we have
k0(Dj1) = kr(Dj2), kr(Dj1) = k0(Dj2), kr−1(Dj1) = kr−1(Dj2), kr−1(Dj1) = kr−1(Dj2).
2 2 2 2
(4)
k (D ) = k (D ) and k (D ) = k (D )
i 11 r−i−1 21 i 12 r−i−1 22
if i ∈ [0,r−1] and r is odd, or if i ∈ [1, r −2]∪[r +1,r−2] and r is even. Moreover, if
2 2 2 2
r is even then
2
k0(D11) = kr−1(D21), kr−1(D11) = k0(D21), kr(D11) = kr−1(D21), kr−1(D11) = kr(D21)
2 2 2 2
(5) ξ(D ) = ξ(D ) = ξ(D ) = ξ(D )
11 12 21 22
(6) Modulo (p−1) we have
r−1
2
s(D )−s(D ) ≡ k (D ) for j = 1,2
j2 j1 i j1
i=0
X
s(D21)−s(D11) ≡ ( (r−4r42−(k12r)(kDr2(1D1)1+1)k+0(D4rk101()D)+11)P+ir2=−21i(r2=i−2−1(i1−)k12r)−ki(2rD−i1(1D) 11) :: 22rr iiss oevdedn
P
(7) k (D ) ∈/ {(0,...,0),(p−1,...,p−1)}
• 11
Lemma 2.6. Sending D = D ⊕D ⊕ D ⊕ D to (n(D ),s(D ),ξ(D )) induces
11 12 21 22 11 11 11
a bijection between the set of isomorphism classes of D-symmetric ´etale (ϕr,Γ)-modules
and S (r).
D
Proof: Again we use Lemma 2.3. For a one-dimensional ´etale (ϕr,Γ)-module D over
k put α(D) = (n(D),s(D),ξ(D)). If D = D ⊕ D ⊕ D ⊕ D is D-symmetric as
E 11 12 21 22
above, then α(D ) is an element of S (r), for all 1 ≤ s,t ≤ 2. Moreover, it is straight-
st D
forward to check ι (α(D )) = α(D ), ι (α(D )) = α(D ), ι (α(D )) = α(D ) and
0 11 12 0 21 22 1 11 21
e
9
ι (α(D )) = α(D ). It follows that the above map is well defined and bijective. (cid:3)
1 12 22
Definition: Wesaythatan´etale(ϕr,Γ)-moduleDoverk isA-symmetricifDadmits
E
a direct sum decomposition D = ⊕r−1D with one-dimensional ´etale (ϕr,Γ)-modules D
i=0 i i
satisfying
j
k (D ) = k (D ), ξ(D ) = ξ(D ), s(D )−s(D ) ≡ k (D ) modulo (p−1)
i j i−j 0 j 0 0 j −i 0
i=1
X
for all i, j (where we understand the sub index in k as the unique representative in
?
[0,r−1] modulo r), and moreover
k (D ) ∈/ {(0,...,0),(p−1,...,p−1)}.
• 0
3 Semiinfinite chamber galleries and the functor D
3.1 Power multiplicative elements in the extended affine Weyl
group
Let G be the group of Q -rational points of a Q -split connected reductive group over Q .
p p p
Fix a maximal Q -split torus T in G, let N(T) be its normalizer in G. Let Φ denote the
p
set of roots of T. For α ∈ Φ let N be the corresponding root subgroup in G. Choose a
α
positive system Φ+ in Φ, let ∆ ⊂ Φ+ be the set of simple roots. Let N = N .
α∈Φ+ α
Let X denote the semi simple Bruhat-Tits building of G, let A denote its apartment
Q
corresponding to T. Our notational and terminological convention is that the facets of
A or X are closed in X (i.e. contain all their faces (the lower dimensional facets at their
boundary)). A chamber is a facet of codimension 0. For a chamber D in A let I be the
D
Iwahori subgroup in G fixing D. We notice that I ∩N = I ∩N .
D α∈Φ+ D α
Fixaspecial vertex x inA, let K bethecorresponding hyperspecial maximal compact
0 Q
open subgroup in G. Let T = T ∩ K and N = N ∩ K. We have the isomorphism
0 0
∼
T/T = X (T) sending ξ ∈ X (T) to the class of ξ(p) ∈ T. Let I ⊂ K be the Iwahori
0 ∗ ∗
subgroup determined by Φ+. [If red : K → K denotes the reduction map onto the
reductive (over F ) quotient K of K, then I = red−1(red(T N )).] Let C ⊂ A be the
p 0 0
chamber fixed by I. Thus I = I and I ∩N = N = N ∩N .
C 0 α∈Φ+ 0 α
We are interested in semiinfinite chamber galleries
Q
(3) C(0),C(1),C(2),C(3),...
in A such that C = C(0) (and thus I = I ) and such that, setting
C(0)
(i)
N = I ∩N,
0 C(i)
10