A Criterion for Integral Structures and Coefficient Systems - IMJ-PRG

Pure and Applied Mathematics Quarterly
Volume 4, Number 4
(Special Issue: In honor of
Jean Pierre, Part 1 of 2 )
1—29, 2008
A Criterion for Integral Structures
and Coefficient Systems on the Tree of P GL(2, F )
Marie-France Vignéras
A Jean-Pierre Serre, avec admiration et amitié
Résumé On donne un critère nécessaire et suffisant pour que l’homologie
de degré 0 d’un système de coefficients GL(2, F )-équivariant, de R-modules libres de type fini, sur l’arbre de P GL(2, F ) soit un R-module libre, lorsque F
est un corps p-adique et R un anneau complet de valuation discrète. Ceci permet de caractériser les représentations lisses p-adiques entières de GL(2, F ), de
construire des structures entières remarquables, définies par un système de coefficients, et par réduction de montrer que les représentations de GL(2, F ) sur un
corps fini de caractéristique p qui se relèvent à la caractéristique 0, sont isomorphes à l’homologie de degré 0 d’un système de coefficients GL(2, F )-équivariant,
d’espaces vectoriels de dimension finie, sur l’arbre. Par exemple, prenons un caractère modérément ramifié p-adique χ1 ⊗ χ2 du tore diagonal T (F ) de GL(2, F ),
tel que χ1 (pF )χ2 (pF ) soit une unité p-adique, qχ1 (pF ) et χ2 (pF ) soient des entiers
p-adiques, pF étant une uniformisante de F et q l’ordre du corps résiduel; alors
la série principale de GL(2, F ) induite lisse non normalisée de χ1 ⊗ χ2 est entière
avec une structure entière remarquable explicite; toute représentation irréductible
supersingulière de GL(2, Qp ) sur un corps fini s’obtient comme réduction d’une
telle structure entière avec χ1 (p) non entier.
Received December 28, 2006.
Marie-France Vignéras
2
Introduction
Let p be a prime number, q a power of p, let F be a local non archimedean
field of characteristic 0 or p, with ring of integers OF and residual field kF = Fq ,
and let G be the group of F -points of a reductive connected F -group. When
E/F is a finite extension, an irreducible locally algebraic E-representation of G
is the tensor product Vsm ⊗E Valg of a smooth one Vsm and of an algebraic one
Valg , uniquely determined ([Prasad] th. 1). The problem of existence of integral
structures in Vsm ⊗E Valg (and their classification modulo commensurability) is
crucial for the p-adic local Langlands correspondence expected to relate p-adic
continuous finite dimensional E-representations of the absolute Galois group GalF
and Banach admissible E-representations of G. The classification of irreducible
representations of G over a finite field k of characteristic p remains a mystery when
G 6= GL(2, Qp ), and the reduction of integral structures is a fundamental open
problem. It is either “obvious” or “very hard” to see if Vsm ⊗E Valg is integral,
and to determine the reduction of an integral structure. It is obvious that a non
trivial algebraic representation Valg is not integral, or that a smooth cuspidal
irreducible representation Vsm with an integral central character is integral.
From now on, G = GL(2, F ).
We fix a local non archimedean field E of characteristic 0 and residual field
kE of characteristic p and we suppose Valg trivial when F is not contained in E,
in particular when the characteristic of F is p. We will present the first local
integrality criterion for Vsm ⊗E Valg , by a purely representation theoretic method,
not relying on the theory of (φ, Γ)-modules as in [Co04], [Co05], [BeBr] or on rigid
analytic geometry as in [Br]. The idea is to realise Vsm ⊗E Valg as the 0-homology
of a G-equivariant coefficient system on the tree [Se77] (an easy generalization
of a general result of Schneider and Stuhler [SS97] for complex finitely generated
smooth representations).
Let X be the Bruhat-Tits tree of P GL(2, F ) with the natural action of G
[Se77]. A fundamental system of G in X consists of an edge σ1 and of a vertex
σo of σ1 . For i = 0, 1, we denote by Ki the stabilizer in G of σi ; the intersection
Ko ∩ K1 has index 2 in K1 , we choose t ∈ K1 not in Ko ∩ K1 , and we denote
by ε the non trivial character of K1 /(Ko ∩ K1 ). Let R be a commutative ring.
A Criterion for Integral Structures ...
3
A G-equivariant coefficient system L of R-modules on X is determined by its
restriction to the vertex σo and to the edge σ1 , i.e. by a diagram
r : L1 → Lo
where r is a R(Ko ∩ K1 )-morphism from a representation of K1 on an R-module
L1 to a representation of Ko on an R-module Lo . The word “diagram” was introduced by Paskunas [Pas] in his beautiful construction of supersingular irreducible
representations of GL(2, F ) on finite fields of characteristic p. The boundary map
from the oriented 1-chains to the 0-chains gives an exact sequence of RG-modules
G
0 → H1 (L) → indG
K1 (L1 ⊗ ε) → indKo Lo → Ho (L) → 0,
where the middle map is the composite of the natural maps indG
K1 (L1 ⊗ ε) →
Ko
G
G
G
G
indKo ∩K1 L1 → indKo ∩K1 Lo = indKo (indKo ∩K1 Lo ) → indKo Lo , and Hi (L) is the
i-th homology of L for i = 0, 1.
The natural RKo -equivariant map wo : Lo → Ho (L) is injective, and the
natural map wo ◦ r : L1 → Lo → Ho (L) is K1 -equivariant (lemma 1.2). If
H1 (L) = 0, then clearly r is injective.
0.1 Basic proposition: local criterion of integrality. Let r : L1 → Lo
be a diagram defining a G-equivariant coefficient system L of R-modules on X .
1) H1 (L) = 0 if and only if r is injective.
2) Suppose that
- R is a complete discrete valuationg ring of fractions field S,
- Lo is a free R-module of finite rank,
- r is injective,
and let L ⊗R S = V, rS = r ⊗R idS : V1 → Vo . Then, the map Ho (L) → Ho (V)
is injective and the R-module Ho (L) is free, when the equivalent conditions are
satisfied :
a) rS (V1 ) ∩ Lo = r(L1 ),
b) the map V1 /L1 → Vo /Lo is injective,
c) r(L1 ) is a direct factor in Lo .
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Marie-France Vignéras
From now on, R is a complete discrete valuationg ring of fractions field S, of
uniformizer x and residue field k = R/xR; for example, the ring OE of fractions
field E, uniformizer pE and residue field kE . An S-representation V of a group
H is called R-integral when V contains an S-basis which generates a R-module
L which is H-stable; one calls L an R-integral structure of V ; the reduction
L = L ⊗R k of L is a k-representation of H.
Remarks (i) If S ′ /S is a finite extension, the integral closure R′ of R in S ′
is complete discrete valuationg ring of fractions field S ′ ([Se] II §2 prop.3); the
diagram L1 → Lo satisfies the properties of 2) if and only if its scalar extension
by ? ⊗R R′ satisfies these properties.
(ii) When the properties of 2) are true, Ho (L) identifies with an R-integral
structure of Ho (V) such that (lemma 1.4.bis)
Ho (L) ∩ Vo = Lo .
(iii) When the properties of 2) are true, the reduction H o (L) = Ho (L) ⊗R k
is the 0-th homology Ho (L) of the G-equivariant coefficient system L = L ⊗R k.
We have the exact sequence of SG-modules:
G
0 → indG
K1 V1 ⊗ δ → indKo Vo → Ho (V) → 0,
of RG-modules:
G
0 → indG
K1 L1 ⊗ δ → indKo Lo → Ho (L) → 0,
of kG-modules:
G
0 → indG
K1 L1 ⊗ δ → indKo Lo → H o (L) → 0.
0.2 Corollary Let V be a G-equivariant coefficient system of S-vector spaces
on X defined by a diagram r : V1 → Vo with r injective. The S-representation
Ho (V) of G is R-integral if and only if there exists an R-integral structure Lo of
the representation Vo of Ko such that L1 = Lo ∩ V1 is stable by t (considering V1
embedded in Vo ).
When this is true, r restricts to a diagram Lo → L1 defining an G-equivariant
coefficient system L of R-modules on X , such Ho (L) identifies with an R-integral
structure of the representation of G on Ho (V).
A Criterion for Integral Structures ...
5
From now on, r is injective and we forget r.
To determine if the S-representation Vo of Ko contains an R-integral structure
Lo such that Lo ∩ V1 is invariant by t, one may proceed inductively as follows.
One reduces easily to Vo = Ko V1 . Let i = 0 or i = 1 identified with an element of Z/2Z. One starts with an arbitrary R-integral structure Mi of the
S-representation Vi of Ki . The RKi+1 -module Mi+1 defined by
- if i = 1, then Mo = Ko M1 ,
- if i = 0, then M1 = (Mo ∩ V1 ) + t(Mo ∩ V1 ),
is an R-integral structure of Vi+1 ; we repeat this construction to get another
R-integral structure z(Mi ) of the S-representation Vi of Ki :
- if i = 1, then z(M1 ) = (Ko M1 ∩ V1 ) + t(Ko M1 ∩ V1 ),
- if i = 0, then z(Mo ) = Ko ((Mo ∩ V1 ) + t(Mo ∩ V1 )).
We call z(Mi ) the first zigzag of Mi ; clearly Mi ⊂ z(Mi ).
0.3 Corollary Let V1 → Vo be an injective diagram as in the corollary 0.2,
Vo = Ko V1 , let i = 0 or i = 1 and let Mi be an arbitrary R-integral structure
of the S-representation Vi of Ki . Then, the S-representation of G on Ho (V) is
R-integral if and only if the sequence of zigzags (z n (Mi ))n≥0 is finite.
From now on, we choose for σo the vertex of the tree associated to the OE module generated by the canonical basis of F 2 and for σ1 the edge between σo
and tσo where
!
0 1
t=
, pF uniformizer of OF .
pF 0
Then Ko = GL(2, OF )Z and K1 =< IZ, t > is the group generated by IZ and t,
where Z is the center of GL(2, F ), isomorphic to F ∗ diagonally embedded, and I
is the Iwahori group of matrices of GL(2, OF ) congruent modulo pF to the upper
triangular group B(Fq ) of GL(2, Fq ).
Set PF = OF !
pF . For an integer e ≥ 1, the e-congruence subgroup K(e) =
!
e
e
1 + PF PF
1 + PFe PFe−1
normalized by Ko is contained in the group I(e) =
PFe
1 + PFe
PFe
1 + PFe
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Marie-France Vignéras
normalized by K1 , generated by K(e) and tK(e)t−1 . The pro-p-Iwahori subgroup
of I is I(1).
0.4 Proposition Let e be an integer ≥ 1, let Vsm be a smooth E-representation
of G generated by its K(e)-invariants and let Valg be an irreducible algebraic Erepresentation of G (hence F ⊂ E if Valg is not trivial).
1) The locally algebraic E-representation V = Vsm ⊗E Valg of G is isomorphic
to the 0-th homology Ho (V) of the coefficient system V associated to the inclusion
I(e)
K(e)
Vsm
⊗E Valg → Vsm
⊗E Valg .
K(e)
2) Suppose that the E-vector space Vsm is finite dimensional. Then, the
representation of G on V is OE -integral if and only if there exists an OE -integral
K(e)
structure Lo of the representation of Ko on Vsm ⊗E Valg such that L1 = Lo ∩
I(e)
(Vsm ⊗E Valg ) is invariant by t. Then the 0-th homology L of the G-equivariant
coefficient system on X defined by the injective diagram L1 → Lo is an OE structure of V .
K(e)
I(e)
Remarks a) By the corollary 0.3, when (Vsm ⊗E Valg ) = Ko (Vsm ⊗E Valg ),
one can suppose Lo = Ko L1 in 2).
K(e)
b) By the lemma 1.4bis, we have Lo = L ∩ (Vsm
⊗E Valg ) in 2).
′ of V
We define the contragredient Ṽ = Ṽsm ⊗E Valg
sm ⊗E Valg as the tensor product of the smooth contragredient Ṽsm of Vsm and of the algebraic contragredient
′ of V .
Valg
alg
0.5 Corollary A finite length locally algebraic E-representation of G is
OE -integral if and only if its contragredient is OE -integral.
0.6 Remark A “moderately ramified” diagram:
- an R-representation Lo of Ko trivial on K(1) with Z acting by a character
ω,
- an R-representation L1 of K1 which is a sum of characters as an I-representation,
I(1)
- an RIZ-inclusion L1 → Lo with image contained in Lo
,
A Criterion for Integral Structures ...
7
is equivalent by “inflation” to a diagram:
- an R-representation Yo of GL(2, Fq ) with Z(Fq ) acting by a character,
- a semi-simple R-representation Y1 of T (Fq ),
N (Fq )
- an RT (Fq )-inclusion Y1 → Yo with image contained in Yo
,
- an operateur τ on Y1 such that:
τ 2 is the multiplication by a scalar a ∈ E ∗ ,
τ permutes the χ-isotypic part and the χs isotypic part of Y1 for any
character χ of T (Fq ).
The action of GL(2, OF ) on Lo is the inflation of Yo , the action of I on L1 is the
inflation of Y1 , the action of t on L1 is given by τ , and a = ω(pF ).
Let V be an R-integral finitely generated S-representation of a group H. There
exists an R-integral structure Lf t of V which is a finitely generated RH-module;
two such R-integral structures Lf t , L′f t are commensurable: there exists a ∈ R
non zero such that aLf t ⊂ L′f t , aL′f t ⊂ Lf t . When the reduction Lf t is a finite
length k-representation of H, the reduction L of an R-integral structure L of V
commensurable to Lf t has finite length and same semi-simplification than Lf t .
See [Vig96] I.9.5 Remarque, and [Vig96] I.9.6).
0.7 Lemma If the reduction Lf t is an irreducible k-representation of H, then
the R-integral structures of V are the multiples of Lf t .
The integral structures constructed in the proposition 0.4 are finitely generated
OE G-modules. We will explicit them when Vsm is a Steinberg representation, or
when V = Vsm is a moderately ramified principal series.
The Steinberg representation Let B = N T be the upper triangular subgroup of G, with unipotent radical N and diagonal torus T . The Steinberg
R-representation StR of G is the R-module of B-left invariant locally constant
functions f : G → R modulo the constant functions, with G acting by right
translations. By [BS] 2.6, we have
StZ ⊗Z R = StR .
Marie-France Vignéras
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The Steinberg representation over any field of characteristic p is irreducible [BL],
[Vig06]. The same definition and the same properties hold when G is replaced
by GL(2, Fq ) [CE] 6.13, ex. 6. The Steinberg R-representation of GL(2, Fq ) will
be denoted by stR . From the lemma 0.7 and [SS91] th. 8, one obtains:
0.8 Proposition The Steinberg R-representation StR of G is the 0-th homology associated to the diagram which is the inflation of the Steinberg RN (F )
representation stR of GL(2, Fq ), the trivial character of T (Fq ) on stR q ≃ R,
N (F )
and the multiplication by −1 on stR q (remark 0.6).
The S-Steinberg representation StS of G is integral, all R-integral structures
are multiple of the Steinberg R-representation StR .
The irreducible algebraic representation Symk of G of dimension k+1 is realized
in the space F [X, Y ]k of homogeneous polynomials in X, Y of degree k ≥ 0 with
coefficients in F ; it has a central character z → z k . We denote by |?| the absolute
value on an algebraic closure F ac of F normalized by |p| = p−1 . Over any finite
k/2
extension E of F [pF ] contained in F ac , the representation
Symk ⊗E | det(?)|k/2
of G has an OE -integral central character. Let us consider the OE -module M1
generated in OE [X, Y ]k by the monomials
(−i+j)/2
X i Y j if i ≤ j and pF
X i Y j if i > j,
and the image φBI in StOE of the characteristic function of BI. One sees that the
OE -integral structure L1 = OE φBI ⊗OE M1 of the representation StI(1) ⊗E Symk ⊗E | det(?)|k/2
of K1 is equal to its zigzag z(L1 ) = Ko L1 ∩ (StI(1) ⊗E Symk ⊗E | det(?)|k/2 ), using
−1/2
1/2
tX = pF uY, tY = pF uX where u ∈ OF∗ , and a small computation.
0.9 Proposition The locally algebraic representation StE ⊗E Symk ⊗E | det(?)|k/2
is OE -integral for any integer k ≥ 0; the 0-th homology of the G-equivariant coefficient system on the tree defined by the diagram
L1 = OE φBI ⊗OE M1 → Lo = Ko L1
is an integral OE -structure.
A Criterion for Integral Structures ...
9
When F = Qp , there are at least four different non trivial proofs of this proposition: three different local proofs have been given by Teitelbaum [T], GrosseKlönne [GK1], Breuil [Br] (with some restrictions), Colmez [Co4], and one can
use modular forms and the cohohomology of modular curves to get a global proof.
Principal series Let χ1 ⊗ χ2 : T → E ∗ be a character of T inflated to B.
The principal series indG
B (χ1 ⊗ χ2 ) is the set of functions f : G → E satisfying
f (hgk) = χ1 (a)χ2 (d)f (g) for all g ∈ G and h ∈ B with diagonal (a, d), and k in
a small open subgroup of G, depending on f , with the group G acting by right
translation. The central character χ1 χ2 is integral if and only if χ1 (pF )χ2 (pF ) ∈
∗ is a unit. When the E-character χ ⊗ χ is moderately ramified, i.e. trivial
OE
1
2
on T (1 + PF ), its restriction to T (OF ) is the inflation of an E-character η1 ⊗ η2
of T (Fq ), and the principal series is the 0-th homology of the G-equivariant
coefficient system defined by the injective diagram
I(1)
K(1)
(indG
→ (indG
B (χ1 ⊗ χ2 ))
B (χ1 ⊗ χ2 ))
which is the inflation of the injective diagram
G(F )
G(F )
(indB(Fqq ) (η1 ⊗ η2 ))N (Fq ) → indB(Fqq ) (η1 ⊗ η2 )
G(F )
with the operator τ on (indB(Fqq ) (η1 ⊗ η2 ))N (Fq ) = Ef1 ⊕ Efs such that
τ f1 = χ1 (pF )fs and τ 2 is the multiplication by χ1 (pF )χ2 (pF ),
where f1 , fs have support B(Fq ), B(Fq )sN (Fq ) and value 1 at id, s. Clearly,
Y1 = OE f1 ⊕ OE χ1 (pF )fs ,
G(F )
is an OE -integral structure of (indB(Fqq ) (η1 ⊗ η2 ))N (Fq ) stable by τ and
Yo = GL(2, Fq )Y1
G(F )
is an OE -integral structure of indB(Fqq ) (η1 ⊗ η2 ). When the central character χ1 χ2
is integral, (Yo , Y1 , τ ) inflates to an injective diagram L1 → Lo = Ko L1 defining
a G-equivariant coefficient system L of free OE -modules of finite rank on X .
An HE (G, I(1))-module is called OE -integral when it contains an E-basis which
generates an OE -module stable by HOE (G, I(1)).
0.10 Proposition When the E-character χ1 ⊗ χ2 is moderately ramified
∗ is a unit, the following properties are equivalent:
and χ1 (pF )χ2 (pF ) ∈ OE
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10
a) the principal series indG
B (χ1 ⊗ χ2 ) is OE -integral,
I(1) is O -integral,
b) the HE (G, I(1))-module (indG
E
B (χ1 ⊗ χ2 ))
c) χ2 (pF ), χ1 (pF )q are integral,
N (Fq )
d) Yo
= Y1 ,
e) the 0-th homology of L is an OE -integral structure L of indG
B (χ1 ⊗ χ2 ).
When they are satisfied, we have LK(1) = Lo , LI(1) = L1 .
When χ1 χ−1
is moderately unramified (trivial on 1 + PF ), one reduces to
2
χ1 ⊗ χ2 moderately ramified by twist by a character. When F = Qp and χ1 χ−1
2
∗
is unramified (trivial on OF ), the equivalence between c) and a) has been proved
by Berger and Breuil [BeBr] using the theory of (φ, Γ)-modules.
0.11 Remarks and questions (i) When χ1 (pF ) is integral, the character
χ1 ⊗χ2 of T is OE -integral and conversely. In this banal case, L is the natural OE integral structure of indG
B (χ1 ⊗χ2 ) of functions with values in OE . Its reduction is
the kE -principal series of G induced from the reduction χ1 ⊗ χ2 of χ1 ⊗ χ2 ; when
χ1 6= χ2 it is irreducible [BL], [Vig04], [Vig06] and the OE -integral structures of
indG
B (χ1 ⊗ χ2 ) are multiple of L, by the lemma 0.7.
When χ1 = χ2 , then indG
B (χ1 ⊗ χ2 ) has length 2; one should be able to prove
that the OE -integral structures of indG
B (χ1 ⊗χ2 ) are OE G-finitely generated; when
F = Qp , compare with [BeBr] 5.4.4 and [Co05] 8.5.
(ii) The module of B is |?|F ⊗ |?|−1
F where |pF |F = 1/q. The contragredient of
−1 −1
G
G −1
indB (χ1 ⊗ χ2 ) is indB (χ1 |?|F ⊗ χ2 |?|F ) ([Vig96] I.5.11), hence the proposition
0.10 and the corollary 0.5 are compatible.
The representation
−1
G
indG
B (χ1 ⊗ χ2 ) ≃ indB (χ1 |?|F ⊗ χ2 |?|F )
is irreducible when χ1 6= χ2 , χ2 |?|2F ; the condition is not symmetric because
the induction is not normalized. This isomorphism is also compatible with the
propositon 0.10.
A Criterion for Integral Structures ...
11
(iii) Theoretically, there is no reason to restrict to the moderately ramified
smooth case, but the computations become harder when the level increases or
when one adds an algebraic part.
(iv) One should see c) as the limit at ∞ of the integrality local criterion. For
k
k/2 , one should replace c) by:
indG
B (χ1 ⊗ χ2 ) ⊗ Sym ⊗| det(?)|
χ2 (pF )q −k/2 , χ1 (pF )q 1−k/2 are integral.
This condition is automatic when the representation is integral; this can be seen
by different, but equivalent, ways either via Hecke algebras or via exponents [E].
The representation of T on the 2-dimensional space of N -coinvariants (Vsm )N of
the smooth part Vsm = | det(?)|k/2 ⊗ indG
B (χ1 ⊗ χ2 ) is semi-simple equal to
| det(?)|k/2 [(χ1 ⊗ χ2 ) ⊕ (χ2 |?|F ⊗ χ1 |?|−1
F )].
N of N -invariants of the algebraic
The character of T on the 1-dimensional space Valg
N is
part Valg = Symk is ?k ⊗ 1. The representation of T on (Vsm )N ⊗E Valg
k/2
k/2
k/2+1
(χ1 ?k |?|F ⊗ χ2 |?|F ) ⊕ (χ2 ?k |?|F
k/2−1
⊗ χ1 |?|F
).
These characters are called the exponents of V . They are integral on the element
!
10
0 pF
which dilates N if and only if χ2 (pF )q −k/2 and χ1 (pF )q 1−k/2 are integral.
Application to k-representations of G.
Let k be a finite field of characteristic p.
The reduction of the integral structure L of a principal series of G induced
from an integral character, constructed in the proposition 0.10 is a principal
series (remark 0.11 (i)), and one deduces:
0.12 Proposition Let µ1 ⊗ µ2 be a k-character of T . The principal series indG
B (µ1 ⊗ µ2 ) is the 0-th homology of the G-equivariant coefficient system
associated to the inclusion
I(1)
K(1)
(indG
→ (indG
,
B (µ1 ⊗ µ2 ))
B (µ1 ⊗ µ2 ))
equivalent by inflation to the data
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12
GL(2,Fq )
- Y o = indB(Fq )
to T (OF ),
N (Fq )
- Yo
(η1 ⊗ η2 ), where η1 ⊗ η2 inflates to the restriction of µ1 ⊗ µ2
N (Fq )
and the operator τ of Y o
such that
τ (f1 ) = µ1 (pF )fs and τ 2 is the multiplication by µ1 (pF )µ2 (pF ).
The reduction of the integral structure L of a principal series of G induced
from a non integral character, constructed in the proposition 0.10 is never a
principal series. The simple HkE (G, I(1))-modules where the action of the center
of HkE (G, I(1)) is as null as possible are called supersingular; they are classified
in [Vig04].
0.13 Proposition Let indG
B (χ1 ⊗ χ2 ) be an OE -integral moderately ramified
principal series of G and let L, Lo , L1 as in the proposition 0.10.
a) χ1 (pF ) is not integral if and only if L1 is a simple supersingular HkE (G, I(1))module.
b) Any simple supersingular HkE (G, I(1))-module arises in a) for some χ1 ⊗χ2 .
c) When L1 = (L)I(1) and L1 is simple supersingular, the kE -representation L
of G is irreducible supersingular.
d) L1 = (L)I(1) when F = Qp .
e) When L1 6= (Lo )I(1) , then L is not admissible and has infinite length.
Remarks (i) When L1 = (L)I(1) , any integral OE -structure of indG
B (χ1 ⊗χ2 )
is a multiple of L by the lemma 0.7.
(ii) The kE -representation L of G is the 0-th homology of the G-equivariant
coefficient system associated to τ , Y 1 → Y o with τ, Y1 , Yo described before the
proposition 0.10. The k-representation Y o = GL(2, Fq )Y 1 has the same semiGL(2,F )
simplification than the reduction W of indB(Fq ) q (η1 ⊗ η2 ). The inclusion
Y 1 ⊂ (Y o )N (Fq )
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13
is an equality when the length of W is 2; this is the case when q = p or when
η1 = η 2 .
From [Vig04], the map V → V I(1) is a bijection between the irreducible krepresentations of GL(2, Qp ) and the simple Hk (G, I(1))-modules. The proposition 0.13 implies:
0.14 Corollary When F = Qp , any irreducible representation of G over a
finite field of characteristic p, with a central character, is the reduction of an
integral principal series.
The results on the integral structures were presented in 2005 in Tel-Aviv and in
Montreal; the application to k-representations was presented in 2006 in Luminy.
At the Montreal conference, Elmar Grosse-Klönne suggested that the function i
on the set of vertices of the Bruhat-Tits building of P GL(n, F ) that he invented
[GK2] could be the key of a generalization of the local integrality criterion to
n ≥ 2.
1 The tree [Se77] The vertices of the tree X of P GL(2, F ) are the similarity
classes [L] of OF -lattices L in a 2-dimensional F -vector space. Two vertices zo , z1
are related by an edge {zo , z1 } when they admit representatives Lo , L1 such that
pF Lo ⊂ L1 ⊂ Lo . The vertex σo := [OF v1 + OF v2 ] for the canonical basis v1 , v2 of
F 2 -vector space, will be the origin of X . The group G = GL(2, F ) acts naturally
on the tree. We have tσo = [OF v1 + OF pF v2 ]. We denote σ1 = {σo , tσo } the edge
between from σo and tσo . The two orientations of the edge σ1 are permuted by t.
The element t normalizes the Iwahori subgroup I and its congruence subgroups
I(e) for e ≥ 1. We choose the combinatorial distance on X (the number of edges
between two vertices). For any integer n ≥ 0, we denote by Sn the sphere of
vertices of distance n to σo and by B(n) the ball of radius n. The action of G
respects the distance.
Coefficient system
Let R be a commutative ring. A G-equivariant coefficient system of R-modules
V on the tree X consists of
Marie-France Vignéras
14
- an R-module Vσ for each simplex σ of X ,
- a restriction linear map rστ : Vτ → Vσ for each edge τ containing the vertex
σ,
- linear maps gσ : Vσ → Vgσ for each simplex σ of X and each g ∈ G, compatible
gτ
with the product of G and with the restriction: (gg′ )σ = gg′ σ gσ′ , gσ rστ = rgσ
gτ .
In particular, the stabilizer of a simplex σ acts on Vσ and the restrictions rστ , rστ ′
are equivariant by the intersection of the stabilizers of the vertices σ, σ ′ of τ .
For any edge τ containing a vertex σ, there exists g ∈ G such that gτ =
σ1 , gσ = σo . This is equivalent to the fact that a G-equivariant coefficient system
is determined by its restriction to the vertex σo and to the edge σ1 , i.e. by a
diagram
r : V1 → Vo
where Vo = Vσo is an R-representation on Ko , V1 = Vσ1 is an R-representation of
K1 and r is an R-linear map which is Ko ∩ K1 -equivariant [Pas].
We denote by X (o) the set of vertices, by X (1) the set of oriented edges (σ, σ ′ )
(with origin σ) and by X1 the set of non oriented edges {σ, σ ′ }.
The R-module Co (V) of oriented 0-chains is the set of functions φ : X (o) →
σ∈X (o) Vσ with finite support such that φ(σ) ∈ Vσ for any vertex σ.
Q
The R-module C1 (V) of oriented 1-chains is the set of functions ω : X (1) →
′
′
{σ,σ′ }∈X1 V{σ,σ′ } with finite support such that ω(σ, σ ) = −ω(σ , σ) ∈ V{σ,σ′ } for
any edge {σ, σ ′ }.
Q
We denote by n(ω) = n the unique integer ≥ 1 such that the support of ω is
contained in the ball Bn and not in Bn−1 .
The boundary ∂ is the R-linear map sending an oriented 1-chain ω supported
on one edge τ = {σ, σ ′ } to the 0-chain ∂ω supported on the vertices σ, σ ′ , with
∂ω(σ) = rστ ω(σ, σ ′ ),
∂ω(σ ′ ) = rστ ′ ω(σ ′ , σ).
The key of the basic proposition is :
For any vertex σ ∈ Sn with n ≥ 1, the neighbours of σ belongs to Sn+1 except
one which belongs to Sn−1 . In particular there is a unique edge τ = τσ starting
A Criterion for Integral Structures ...
15
from σ and pointing toward σo , and
(key)
∂ω(σ) = rστσ ω(τσ ) for σ ∈ Sn(ω) .
The group G acts on the R-module of oriented ∗-chains by
(gω)(gσ) = g(ω(σ))
for any g ∈ G and any oriented ∗-chain ω. The boundary ∂ is G-equivariant.
The representation of G on Co (V) and on C1 (V) are isomorphic to the compactly
G
∗
induced representations indG
Ko Vo and indK1 (V1 ⊗ ε) where ε : K1 → R is the
R-character of K1 trivial on Ko ∩ K1 such that ε(t) = −1. The 0-homology
Ho (V) =
Co (V)
∂C1 (V)
is an R-representation of G. The same is true with the 1-homology H1 (V) =
Ker ∂.
We identify naturally vo ∈ Vo with a 0-chain with support on the single vertex
σo . This induces a Ko -equivariant linear map
wo : Vo → Ho (X , V)
that we can “restrict” to V1 to get a Ko ∩ K1 -equivariant map
wo rσσo1 : V1 → Vo → Ho (X , V).
1.2 Lemma The map wo is injective and the map wo rσσo1 is K1 -equivariant.
Proof. No 1-chain is supported on a single vertex and the key formula gives
the injectivity. For v1 ∈ V1 , if ω is the oriented 1-chain ω with support the edge
σ1 such that ω(σo , tσo ) = tv1 , we have
∂(ω) = ωo rσσo1 (tv1 ) − t(ωo rσσo1 (v1 )).
This gives the K1 -equivariance.
When the map
rσσo1 : V1 → Vo
16
Marie-France Vignéras
is injective, then the map rστ is injective for any vertex σ and any edge τ containing
σ.
1.3 Lemma ∂ is injective if the map rσσo1 is injective.
Proof. Let ω 6= 0. Let σ ∈ Sn(ω) such that the edge τσ (starting from σ
and pointing towards the origin) belongs to the support of ω. As rστ is injective,
∂(ω)(σ) = rστ ω(τσ ) does not vanish.
We suppose from now on that the map rσσo1 : V1 → Vo is injective.
Let φ be a non zero 0-chain not supported on the origin. There exists a unique
integer n(φ) ≥ 1 such that the support of φ is contained in the ball Bn(φ) and not
in Bn(φ)−1 . Let σ ∈ Sn(φ) such that φ(σ) 6= 0; let τ be the edge pointing towards
the origin, starting from σ. If φ(s) belongs to rστ Vτ , then by the key formula, we
can find an oriented 1-chain ω supported on the edge τ such that φ − ∂ω vanishes
at σ. Obviously φ − ∂ω is equal to φ outside σ ∪ Sn(φ)−1 .
When the R-module image of V1 in Vo has a supplementary Wo
Vo = Wo ⊕ rσσo1 (V1 ),
the image of Vτ in Vσ has a (non canonical) supplementary Wσ ,
Vσ = Wσ ⊕ rστ (Vτ ).
We can find an oriented 1-chain ω supported on τσ such that (φ − ∂ω)(σ) ∈ Wσ .
Hence any non zero element of Ho (V) has a representative φ either supported on
the origin, or such that n(φ) ≥ 1 and φ(σ) ∈ Wσ for any σ ∈ Sn(φ) .
1.4 Proof of the basic proposition 0.1.
1) is a particular case of the lemma 1.2.
2) Equivalence of the properties a), b), c). It is obvious that V1 ∩ Lo = L1 is
equivalent to: the kernel of V1 → Vo /Lo is L1 , is equivalent to: the quotient of
Lo by L1 is torsion free, and is equivalent to L1 is a direct factor of Lo because
Lo is a free module of finite rank over the principal ring R.
A Criterion for Integral Structures ...
17
The injectivity Ho (L) → Ho (V) results from the condition b), the lemma 1.3
applied to the coefficient system V/L, the exactitude of the sequence H1 (V/L) →
Ho (L) → Ho (V).
One can also prove the injectivity as follows. If ω ∈ C1 (V) and ∂ω ∈ Co (L) then
τ?
r? ω(τ? ) ∈ L? for any vertex ? ∈ Sn(ω) . By the condition a), and the injectivity
of r?τ , we have ω(τ? ) ∈ Lτ? . One repeats the same argument to ω − ωext where
ωext is the restriction of ω to the oriented edges between Sn(w)−1 and Sn(w) .
The R-module Ho (L) embedded in the countable S-vector space Ho (V) is Rfree if and only if it contains no line because R is a local complete principal ring.
Let v be a non zero element of Ho (L). We choose
- a representative φ of v such that φ is supported on σo or such that n(φ) ≥ 1
and φ(σ) ∈ Wσ for any σ ∈ Sn(φ) ,
- a vertex σ ∈ Sn(φ) such that φ(σ) 6= 0,
- an integer n ≥ 1 such that φ(σ) does not belong to xn Lσ for the chosen vertex
σ ∈ Sn(φ) .
Suppose that the line Sv is contained in Ho (L). Then, there exists an oriented
1-cocyle ω ∈ C1 (L) such that (φ + ∂(ω))(?) belongs to xn Ls for any vertex
?. If n(ω) > n(φ), the key formula implies that ω(τ? ) belongs to pn L(τ ) for
any vertex ? ∈ Sn(ω) , because the condition a) and the injectivity of rτ∗ imply
rτ? Lτ ∩ xn L? = rτ? (xn Lτ ). The oriented 1-cocycle ω − ωext has the same property
than ω. By decreasing induction on n(ω) we may suppose n(ω) ≤ n(φ). If φ
is supported on σo , then ω = 0 and φ(σo ) ∈ xn Lo which is false. Otherwise,
φ(?) + ω(τ? ) ∈ xn L? for any ? ∈ Sn(φ) . As φ(?) ∈ W? and ω(τ? ) ∈ r?τ? (Vτ? ), this
is impossible.
1.4bis Lemma Let φ be a 0-chain with support on the single vertex σo and
let ω be an oriented 1-chain such that φ + ∂(ω) is integral. Then φ is integral.
Proof. As n(ω) ≥ 1, the restriction of ω on Sn(ω) is integral by the key formula.
By a decreasing induction on n(ω), φ is integral.
1.5 Proof of the corollary 0.2
18
Marie-France Vignéras
Sufficient. When Lo is an R-integral structure of Vo such that L1 = Lo ∩ V1
is stable by t, then L1 is an R-integral structure of V1 ; the map r induces a
diagram L1 → Lo and a coefficient system L on X such that V = L ⊗R S. By
the proposition 0.1, Ho (V) is R-integral.
Necessary. Suppose that L is an R-integral structure of Ho (V). We apply
the lemma 1.2. The inverse image Lo of wo (Vo ) ∩ L in Vo by the injective Ko equivariant map wo is an R-integral structure of the representation of Ko on
Vo , and the inverse image L1 of wo (V1 ) ∩ L is an R-integral structure of the
representation of K1 on V1 , equal to Lo ∩ V1 (of course stable by t).
Apply the proposition 0.1 for the integral structure.
1.6 Proof of the corollary 0.3
1) Sufficient. When the sequence of zigzags of some R-integral structure of Vo
or of V1 is finite, there exists for i = 0 and for i = 1, an R-integral structure Mi
of Vi equal to its first zizgag z(Mi ) = Mi .
- When z(M1 ) = M1 , then Lo = Ko M1 satisfies M1 = Lo ∩ V1 is stable by t
and by the corollary 0.2, Ho (V) is R-integral.
- When z(Mo ) = Mo , then Lo = Mo satisfies Lo ∩ V1 is stable by t and by the
corollary 0.2, Ho (V) is R-integral.
2) Necessary. When Ho (V) is R-integral, there exists an R-integral structure
Lo of Vo such that L1 = V1 ∩ Lo is t-invariant by the proposition 0.2.
- Let M1 be an R-integral structure of V1 . Replacing L by a multiple, we
suppose M1 ⊂ L1 . Then Ko M1 ⊂ Lo and z(M1 ) ⊂ L1 . The index [L1 : M1 ]
is finite and the sequence of zizgags of M1 is increasing; hence the sequence of
zizgags is finite.
- Let Mo be an R-integral structure of Vo . We apply the above argument to
M1 = (V1 ∩ Mo ) + t(V1 ∩ Mo ); the sequence of zizgags of M1 is finite, and also
the sequence of zizgags of Mo .
1.7 Proof of the proposition 0.4
1)The exactness of the sequence
I(e)
G
K(e)
0 → indG
⊗E Valg ) → Vsm ⊗E Valg → 0
K1 (Vsm ⊗E Valg ⊗E ε) → indK1 (Vsm
A Criterion for Integral Structures ...
19
follows from the following facts.
For any complex smooth representation Vsm of G generated by its K(e)invariants vectors, the natural sequence (where the middle map is the boundary)
I(e)
G
K(e)
0 → indG
K1 (Vsm ⊗ ε) → indK1 (Vsm ) → Vsm → 0
is exact [SS97] II.3.1). This remains true for an E-representation Vsm , because
the scalar extension ⊗E C commutes with the invariants by an open compact subgroup and with the compact induction from an open subgroup; a short sequence
of E-vector spaces is exact if and only if its scalar extension to C is exact. The
tensor product by ⊗E Valg of an exact sequence of EG-representations remains
exact and ⊗E Valg commutes with the compact induction from an open subgroup.
K(e)
2) When Vsm
is finite dimensional, we apply the corollary 0.2.
1.8 Proof of the corollary 0.5
(i) Let Vsm be a smooth E-representation of G of finite length; there exists an
integer e ≥ 1 such that each non zero irreducible subquotient of Vsm contains a
K(e)
non zero K(e)-invariant vector. If Vsm 6= 0 then (Ṽsm )K(e) isomorphic to the
K(e)
dual of Vsm is not 0; the irreducible subquotients of the contragredient Ṽsm
are the contragredients of the irreducible subquotients of Vsm . Hence Vsm and
K(e)
Ṽsm are generated by their K(e)-invariants. The E-vector space Vsm is finite
dimensional because a finite length smooth E-representation of G is admissible.
All these facts are known for complex representations [Vig96]. They remain true
for E-representations because Vsm ⊗E C has finite length [Vig96] II.43.c, and
? ⊗E C commutes with the K(e)-invariant functor.
(ii) Suppose that V = Vsm ⊗E Valg is OE -integral. We choose an OE -integral
K(e)
structure Lo of the representation of Ko on Vsm ⊗E Valg such that L1 = Lo ∩
I(e)
(Vsm ⊗E Valg ) is t-stable (proposition 0.4). Consider the algebraic dual L′i =
′ . Then L′ is an O -integral
HomOE (Li , OE ) for i = 0, 1 and Ṽ = Ṽsm ⊗E Valg
E
o
structure of the representation of Ko on the dual
K(e)
K(e) ′
′
(Vsm
⊗E Valg )′ ≃ (Vsm
) ⊗E (Valg )′ ≃ (Ṽsm )K(e) ⊗E Valg
I(e)
and L′1 is an OE -integral structure of the representation of K1 on (Vsm ⊗E
I(e) ⊗ V ′ ) send L
′ . The linear forms in L′ ∩ ((Ṽ
Valg )′ ≃ (Ṽsm )I(e) ⊗E Valg
1
sm )
E alg
o
′
′
in OE . Conversely, the restriction to L1 gives a surjective map Lo → L1 because
Marie-France Vignéras
20
the OE -module L1 is a direct factor of Lo , hence an isomorphism
′
) ≃ L′1 .
L′o ∩ ((Ṽsm )I(e) ⊗E Valg
By the proposition 0.4, Ṽ is OE -integral.
1.9 Proof of the lemma 0.7
Let L be an R-integral structure of V which is different from Lf t . Taking a
multiple of Lf t , we reduce to Lf t ⊂ L and Lf t not contained in xL. We have the
inclusions
xLf t ⊂ (xL ∩ Lf t ) ⊂ Lf t .
The irreducibility of Lf t /xLf t implies xL ∩ Lf t = xLf t , equivalent to L = Lf t
because ∪n≥0 x−n Lf t = V .
2 The Steinberg representation
The proposition 0.8 results from the lemma 0.7 and from the remarkable properties of the Steinberg R-representation StR of G that we recall.
2.1 The R-Steinberg representation stR = stZ ⊗Z R of GL(2, Fq ) is a free
K(1)
is the inflation of stR
R-module of rank q. The representation of Ko on StR
because G = BGL(2, OF ). We have
I(1)
StR
K(1)
= RφBI , RKo φBI = StR
,
tφBI = −φBI ,
where φBI is the characteristic function of BI modulo the constants, by the same
proof of [BL] lemma 26 for the first equality, because the characteristic function
G(F )
of B(Fq ) generates indB(Fqq ) 1R for the second equality (see also the lemma 2.7),
and because φBI (t) = −φBsI (t) = −1 for the third equality.
2.2 The Steinberg R-representation StR of G is the highest cohomology with
compact supports of the tree by [BS] 5.6, and by [SS91] cor. 17,
StR = R[G/I] ⊗HR (G,I) sign .
where sign is the character of the Hecke algebra HR (G, I) of the Iwahori subgroup
I(1)
I on StIR = StR .
A Criterion for Integral Structures ...
21
2.3 StR is the 0-th homology of the G-coefficient system associated to the
inclusion
I(1)
StR
K(1)
→ StR
by [SS91] th. 8.
2.4 StR = StZ ⊗Z R is an R-free module, isomorphic as an R-representation
of B to Cc∞ (N, R), where N acts by translations and T by conjugation, by the
map [BS] 3.7:
!
01
f → φf (n) = f (n) − f (sn) for n ∈ N and s =
.
10
K(1)
One can check that the image of StR
GL(2, OF ) and N (1) = N ∩ K(1).
is Cc∞ (N (0), R)N (1) where N (0) = N ∩
2.5
When R is
! a field of characteristic p, the action of the monoid generated
pF 0
by N and
on Cc∞ (N, R) is irreducible [Vig06]. The unique projective
0 1
irreducible R-representation of GL(2, Fq ) is stR [CE] 6.12.
!
10
One embeds Fq in OF by the Teichmüller map. Set vx =
. A system
x1
of representatives of Ko /ZI ≃ GL(2, Fq )/B is s, (vx )x∈Fq because GL(2, Fq ) =
B(Fq ) ∪ N (Fq )sB(Fq ). If M is an RZI-module, then
X
vx M.
Ko M = sM +
x∈Fq
2.6 Lemma sφBI +
P
x∈Fq
vx φBI = 0. When R is a field, the q elements
K(1)
sφBI , vx φBI (x ∈ F∗q ) form an R-basis of StR
K(1)
Proof. RKo φBI = StR
and
o
StK
R
= 0.
.
(2.3), the sum sφBI +
P
x∈Fq
vx φBI is Ko -invariant
2.7 Proof of the integrality of St ⊗Sk (proposition 0.9).
Marie-France Vignéras
22
Let k ≥ 1 be an integer, and L1 = OE φBI ⊗OE M1 as in the proposition 0.9.
Let N be the intersection of sM1 with ∩x∈F∗q vx M1 . By the lemma 2.6,
Ko L1 = (sφBI ⊗ sM1 ) +
X
(vx φBI ⊗ vx M1 ).
x∈F∗q
and the first zizgag of L1 is
I(1)
z(L1 ) = Ko L1 ∩ (StE
⊗E Symk ⊗E | det(?)|k/2 ) = OE φBI ⊗OE (M1 + N ).
The key of the proof is that N = Lo . A basis of the OE -module Lo is X i Y j
(−j+i)/2 i j
X Y if
for i, j ∈ N, i + j = k; a basis of sM1 is X i Y j if i ≥ j and pF
i < j for i, j ∈ N, i + j = k; a basis of vx M1 is (X + xY )i Y j if i ≤ j and
(−i+j)/2
(X + xY )i Y j if i > j for i, j ∈ N, i + j = k. Suppose that
pF
X
X
ci,j X i Y j =
di,j (x)(X + xY )i Y j (ci,j , di,j (x) ∈ E, x ∈ F∗q )
i+j=k
i+j=k
belongs to N . Modulo Lo , we can forget the ci,j with i ≥ j and di,j (x) with i ≤ j.
We get
X
X
di,j (x)(X + xY )i Y j mod Lo .
ci,j X i Y j ≡
i<j
j<i
When k = 2u is even, X i does not appear on the left side if i ≥ u. By decreasing
induction on i, we can show that dk,o (x), dk−1,1 (x) . . . , du,u (x) ∈ OE . When
k = 2u + 1 is odd, X i does not appear on the left side if i ≥ u + 1, and we can
show that the di,j (x) for j < i belong to OE . Hence N = Lo .
As Lo ⊂ M1 , one deduces from N = Lo that L1 is equal to its first zigzag
z(L1 ). We apply the corollaries 0.3 and 0.2 and the proposition 0.9 is proved.
3 Principal series. Proof of proposition 0.10
Let χ1 ⊗ χ2 : T → S ∗ be a moderately ramified S-character of T such that
χ1 (pF )χ2 (pF ) ∈ R∗ . We set
θ = η1−1 η2
where η? is the restriction of χ? to OF∗ for ? = 1, 2; clearly η? , θ are R-characters.
A moderately ramified character of OF∗ is the inflation of a character of F∗q , that
we denote by the same letter.
A Criterion for Integral Structures ...
23
As G = BGL(2, OF ), the restriction to GL(2, OF ) of indG
B (χ1 ⊗ χ2 ) is isomorphic to
GL(2,O )
indB(OF ) F (η1 ⊗ η2 ).
K(1) is the inflation of the
The representation of GL(2, OF ) on (indG
B (χ1 ⊗ χ2 ))
principal series
GL(2,Fq )
indB(Fq )
(η1 ⊗ η2 ).
A system of representatives of B(OF )\GL(2, OF )/K(1) ≃ B(Fq )\GL(2, Fq ) is
!
1x
1, sux for x ∈ Fq , ux =
.
01
by the decomposition GL(2, Fq ) = B(Fq ) ∪ B(Fq )sN (Fq ). We use the Teichmüller embedding Fq → OF . Let Lo be the natural integral R-structure
GL(2,O )
of the representation of Ko on (indB(OF ) F (η1 ⊗ η2 ))K(1) , given by the functions
with values in R. We denote by fg the function in Lo with support B(OF )gK(1)
such that fg (g) = 1. An R-basis of Lo is {f1 , (fsux )x∈Fq }. The OE Ko -module Lo
is cyclic generated by f1 because
ux sf1 = fsu−x
for x ∈ Fq .
Modulo the first congruence group K(1), the pro-p-Iwahori I(1) is represented
I(1)
by (ux )x∈Fq . A basis of Lo is
X
fsux .
f1 ,
x∈Fq
!
pF 0
It is convenient to write t = sh = shss where h =
. It is obvious that
0 1
tf1 (1) = f1 (t) = 0, tf1 (s) = f1 (st) = χ(h) = χ1 (pF ), hence
X
fsux
tf1 = χ1 (pF )
x∈Fq
and because t2 = pF id,
t
X
fsux = χ2 (pF )f1 .
I(1)
is equal to
x∈Fq
The R-module L1 =
I(1)
Lo
+ tLo
L1 = (R + χ2 (pF )R)f1 ⊕ (R + χ1 (pF )R)
X
x∈Fq
fsux .
Marie-France Vignéras
24
I(1)
We see that the module Lo is stable by t if and only if χ1 (pF ) and χ2 (pF )
belong to R (hence units). Otherwise, as Lo = RKo f1 , the zigzag z(Lo ) = Ko L1
contains χ2 (pF )Lo ; if the sequence of zigzags (z n (Lo ))n≥0 is finite, then χ2 (pF )
is integral. By the corollary 0.3, we may suppose χ2 (pF ) integral and χ1 (pF ) not
integral. Then
X
X
fsuc .
fsuc , Ko L1 = Lo + χ1 (pF )RKo
L1 = Rf1 + χ1 (pF )R
c∈Fq
c∈Fq
A system of representatives of Ko /ZI(1) ≃ GL(2, Fq )/Z(Fq )N (Fq ) is
!
λ0
∗
{dλ , dλ ux s for all x ∈ Fq , λ ∈ Fq }, dλ =
.
01
P
Hence the RKo -module Mo generated by c∈Fq fsuc is the R-module generated
by
X
X
fsuc , for all x ∈ Fq , λ ∈ F∗q .
fsuc , dλ ux s
dλ
c∈Fq
c∈Fq
We compute Mo . As suc dλ = sdλ ssuλ−1 c we have
dλ f1 = η1 (λ)f1 ,
dλ fsuc = η2 (λ)fsuλc .
Hence
dλ
X
fsuc = η2 (λ)
X
fsuc .
c∈Fq
c∈Fq
P
P
As η2 (λ) is a unit in R, we have Rdλ c∈Fq fsuc = R c∈Fq fsuc . As suc ux s =
vc+x , and
!
−1/x 1
vx =
su1/x when x 6= 0,
0
x
we have
ux sfs = f1 , ux sfsuc = χ1 (−1)θ(x + c′ )fsuc′
if c ∈ F∗q with x + c′ = c−1 . Hence
ux s
X
fsuc = Fx
c∈Fq
where
Fx = f1 + χ1 (−1)
X
c∈Fq
θ(x + c)fsuc
A Criterion for Integral Structures ...
25
(a character of F∗q is extended to a function on Fq vanishing on 0). As θ = η1 η2−1 ,
we have
X
X
θ(x + c)fsuλc = η1 (λ)Fλx .
fsuc = η1 (λ)f1 + η2 (λ)χ1 (−1)
dλ ux s
c∈Fq
c∈Fq
P
As η1 (λ) is a unit in R, we have Rdλ ux s c∈Fq fsuc = RFλx . We deduce that Mo
is the R-module generated by
X
fsuc , (Fx )x∈Fq .
c∈Fq
P
The sum x∈Fq Fx is qf1 + (q − 1) c∈Fq fsuc if θ is the trivial character, and
qf1 if θ is not trivial. Hence Mo contains qf1 ; beeing Ko -stable, Mo contains qLo .
The zizgag z(Lo ) = Ko L1 = Lo + χ1 (pF )Mo contains qχ1 (pF )Lo . If the sequence
of zigzags (z n (Lo ))n≥0 is finite, then qχ1 (pF ) is integral.
P
By the corollary 0.3, it remains to consider the case χ1 (pF ) not integral and
qχ1 (pF ) integral. We compute (Ko L1 )I(1) ; as Lo + χ1 (pF )Mo is the R-module
generated by
X
fsuc , (χ1 (pF )Fx )x∈Fq ,
Lo , χ1 (pF )
c∈Fq
(Ko L1 )I(1) is the R-module generated by L1 and by
X
X
fsuc
ax f1 + χ1 (−1)s(0)
c∈∈Fq
x∈Fq
for all functions a? : Fq → χ1 (pF )R such that the function s(?) : Fq → χ1 (pF )R
defined by
X
ax θ(? + x)
s(?) =
x∈Fq
P
is constant modulo R. As L1 contains χ1 (pF ) c∈∈Fq fsuc , the R-module (Ko L1 )I(1)
P
is generated by L1 and by f1 x∈Fq ax for all a? as above. Using once again that
P
Lo = RKo f1 , if the sequence of zizgags z n (Lo )n≥0 is finite then x∈Fq ax ∈ R.
P
Conversely, x∈Fq ax ∈ R for all a? as above implies z(L1 ) = L1 .
P
3.1 Lemma
x∈Fq ax ∈ R for each a? : Fq → χ1 (pF )R such that s(?) is
constant modulo R.
26
Marie-France Vignéras
P
Proof. When the character θ is trivial, we have s(?) + a? = c∈Fq ac . If s(?)
is constant modulo R, then a? is constant modulo R; if moreover a? ∈ χ1 (pF )R,
P
we have x∈Fq ax ∈ qχ1 (pF )R and qχ1 (pF ) is integral.
When θ is not the trivial character, the proof is more complicated. We use
Fourier transform; we need to suppose that R contains a p-th root of 1 for the
existence a non trivial character ψ : Fq → R to define the Fourier transform
X
ψ(x?)f (x)
fˆ(?) =
x∈Fq
of a function f : Fq → S. We denote by R the space of integral functions f :
Fq → R, by R̂ the image of R by Fourier transform, by δo ∈ R the characteristic
function of 0 and by ∆ ∈ R the constant function ∆(?) = 1. The remarkable
properties of the Fourier transform give
ˆ
ˆ = qδo , δ̂o = ∆, θ̂(x) is a Gauss sum when x ∈ F∗q , and θ̂(0) = 0;
fˆ = qf, ∆
ˆ )(x) = qθ(−1) for x ∈ F∗ ;
θ̂(x)(θ −1
q
the Fourier transform of a convolution product f ∗ g for f, g : Fq → S is the
product of the Fourier transforms
X
f (y)g(z), f ˆ∗ g = fˆĝ.
f ∗ g(x) =
y,z∈Fq ,y+z=x
The lemma says that â(0) ∈ R for all a ∈ χ1 (pF )R such that a ∗ θ ∈ R∆ + R.
By Fourier transform a ∗ θ ∈ R∆ + R is equivalent to âθ̂ ∈ Rqδo + R̂. Mulˆ ) vanishing only at 0, this is equivalent to qâ = qâ(0)δo + (θ −1
ˆ )φ̂
tiplying by (θ −1
for some φ ∈ R. The function b = qa belongs to R because qχ1 (pF ) ∈ R
ˆ )φ̂. By Fourier transform b = λ∆ + θ −1 ∗ φ with λ =
and b̂ = b̂(0)δo + (θ −1
b(0) − (θ −1 ∗ φ)(0) ∈ R and b̂(0) = qλ. Hence â(0) ∈ R.
We summarize what we proved in the following proposition.
I(1)
3.2 Proposition 1) Lo = L1 and Ko L1 = Lo if and only if χ1 (pF ), χ2 (pF )
belongs to R (hence are units).
2) If the sequence (z n (Lo )n≥0 is finite, then qχ1 (pF ), χ2 (pF ) belong to R.
A Criterion for Integral Structures ...
27
3) When χ1 (pF ) is not integral, but qχ1 (pF ), χ2 (pF ), then (Ko L1 )I(1) = L1 if
θ is trivial, or if R contains a p-root of 1.
We prove now the proposition 0.10. By [Vig04 prop. 4.4], the properties b), c)
are equivalent. The proposition 3.2 implies that the properties a), c), d), e) are
equivalent, using the propositions 0.1 and 0.4, the remark (i) to the proposition
0.1 (as E has characteristic 0, a finite extension contains a p-root of 1), the
corollary 0.3 and the remark 0.6. One gets also that Lo = Ko L1 . By the lemma
I(1)
1.4bis, Lo = LK(1) for L as in e). Hence LI(1) = Lo = L1 .
4 k-representations
4.1 Proof of the proposition 0.12
Let µ1 ⊗ µ2 : T → k∗ be a continuous character. There exists a moderately
ramified continuous character χ1 ⊗χ2 : T → R∗ lifting µ1 ⊗µ2 . By the proposition
0.10 and the remark 0.11 (i), indG
B (µ1 ⊗ µ2 ) is the 0-homology of the diagram
N (Fq )
associated to (Yo , Yo
, τ ) given in the proposition 0.12.
4.2 Proof of the proposition 0.13
a), b), c) [Vig04], d) follows from the lemma 4.3. e) follows from the lemma
4.4 and the inclusions of k-representations
indG
K1
indG
(Lo )I(1)
K o Lo
⊂L=
.
G
L1
indK1 L1
4.3 Lemma A k-representation V of G = GL(2, Qp ) generated by a simple
Hk (G, I(1))-submodule M of V I(1) , is irreducible and M = V I(1) .
Proof. By [Ollivier], the k-representation of G
M ⊗Hk (G,I(1)) indG
I(1) 1k
is irreducible of I(1)-invariants M ≃ M ⊗ 1, and V is a quotient of M ⊗Hk (G,I(1))
indG
I(1) 1k .
28
Marie-France Vignéras
4.4 Lemma The k-representation indG
K1 W has infinite length and is not
admissible, for any non zero smooth k-representation W of K1 with pF acting by
multiplication by λ ∈ k∗
Proof. On may prove it after a scalar extension and a twist by a character of
hence we suppose that k contains F∗q , and that λ = 1. We may also suppose
W irreducible. The group G has infinitely many non isomorphic irreducible krepresentations containing a given irreducible representation W of K1 with pF
acting trivially (this is easily deduced from [Vig04]); they are quotients of indG
K1 W
G
hence indK1 W has infinite length; it is not admissible because K1 \G/I1 is infinite
and each non representation of I(1) has a non zero I(1)-invariant vector.
pZ
F;
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