Acyclicity of Schneider and Stuhler's coefficient systems:another approach in the level 0 case

0.17 MB·English

by  Paul Broussous

Acyclicity of Schneider and Stuhler’s coefficient
systems: another approach in the level 0 case
4
0
Paul Broussous
0
2 UMR 6086 du CNRS et Universit´e de Poitiers

n
T´el´eport 2 - BP 30179
a
J
Bd Pierre et Marie Curie

6
86962 Futuroscope Chasseneuil

]
T France
R
Email: [email protected]
.
h
t
a February 1, 2008
m
[

2
v Abstract
3
8
1 Let F be a non archimedean local field and G be the locally profi-
2
nite group GL(N,F), N ≥ 1. We denote by X the Bruhat-Tits build-
1
3 ing of G. For all smooth complex representation V of G and for all
0 level n > 1, Schneider and Stuhler have constructed a coefficient sys-
h/ tem C = C(V,n) on the simplicial complex X. They proved that
at if V is generated by its fixed vectors under the principal congruence
m subgroup of level n, then the augmented complex Cor(X,C) → V of
•
: oriented chains of X with coefficients in C is a resolution of V in the
v
i category of smooth complex representations of G. In this paper we
X
give another proof of this result, in the level 0 case, and assuming
ar moreover that V is generated by its fixed vectors under an Iwahori
subgroup I of G. Here “level 0” refers to Bushnell and Kutzko’s ter-
minology, that is to the case n= 1+0. Our approach is different. We
strongly use the fact that the trivial character of I is a type in the
sense of Bushnell and Kutzko.
Keywords: Simplicial complex, affine building, p-adic groups, smooth
representation, coefficient system, resolution, homological algebra.
AMS Classification: 22E50, 18G10.
1
1 Introduction
Let F be a locally compact non-archimedean field and G be the group
GL(N,F), where N > 1 is a fixed integer. We denote by X the semisimple
affine building of G. We let R(G) denote the category of smooth complex
representations of G. For any simplex σ of X, we denote by U the pro-
σ
unipotent radical of the (compact) parahoric subgroup P attached to σ. We
σ
fix a representation V ∈ R(G) and we assume that it is generated by its fixed
vector under some Iwahori subgroup I. In [6] Schneider and Stuhler define
a coefficient system C = (V ) , where V = VUσ for all simplex σ of X. The
σ σ σ
central result of loc. cit. is that the augmented complex of oriented chains
of X with coefficients in C
0 → Cor (X,C)→∂ ···→∂ Cor(X,C)→ǫ V (1)
N−1 0
is a resolution of V in R(G). Of course the result of [6] is more general
and is concerned with representations of any level.
The aim of this paper is to give another proof of the acyclicity of the
homological complex (1). Our proof proceeds as follows. Let R (G) be the
(I,1)
full subcategory of R(G) whose representations are generated by their fixed
vectors under I. Let H(I,1) be the Hecke algebra of the trivial character of
I. Then since (I,1) is a type for G (see [2], [3]), we have an equivalence of
categories
R (G) −→ Mod(H(I,1))
(I,1)
W 7→ WI
where Mod(H(I,1)) is the categoy of left unital modules over H(I,1). We
then show that the complex (1) is actually a complex in R (G). Applying
(I,1)
the equivalence of categories to (1), we show that we obtain a complex iso-
morphic to the chain complex of the standard apartment of X with constant
coefficients in VI. Since this apartment is contractible, this latter complex is
exact.
The paper is organized as follows. In §7 we compute the image of the
complex (1) under the equivalence of categories. In §8 we show how this
image is related to the chain complex of the standard apartment with coef-
ficients in VI. Proofs are based on some geometrical lemmas (cf. §5) and on
a key lemma on idempotents of the Hecke algebra of G (cf. §6).
I thank Guy Henniart and Peter Schneider for stimulating discussions.
The proof of the main result of §6 is due to Jean-Franc¸ois Dat.
2
2 Notation
We keep the notation of the introduction. We shall denote by Y the set of
q
q-dimensional simplices of a simplicial complex Y, q ∈ Z . We set
>0
Go = {g ∈ G ; det(g) ∈ o×} ,
F
where o denotes the ring of integers on F. We fix a maximal split torus T of
F
G and denote by A the corresponding appartement of X. We fix a chamber
C of A and write
o
I = {g ∈ Go ; gC = C }
o o
for the Iwahori subgroup attached to C . The subgroup I together with
o
No = {g ∈ Go ; gTg−1 = T}
formanaffineBN-pairinGo. The corresponding affineWeylgroupisWaff =
No/To, where To ⊂ T is the maximal compact subgroup of T. Recall that
X is the geometric realization of the combinatorial building of the BN-pair
(I,No).
We fix a labelling λ : X → ∆ (cf. [1], page 29), where ∆ is the
N−1 N−1
standard (N−1)-simplex. The action of Go is label preserving. The labelling
λ gives rise to an orientation of the simplices of X and to incidence numbers
[σ : τ] , τ ⊂ σ, dimσ = dimτ +1. The action of Go preserves the orientation
and the incidence numbers.
Finally we fix a smooth representation V in the subcategory R (G) of
(I,1)
R(G). We have a G-equivariant coefficient system
C = (V )
σ σ
on X, where V = VUσ, for all simplex σ. Here the restriction maps rσ :
σ τ
V → V , for τ ⊂ σ, are the inclusion maps.
σ τ
3 The chain complexes
The coefficient system C gives rise to the augmented complex of oriented
chains of X with coefficients in C:
0 → Cor (X,C)→∂ ···→∂ Cor(X,C)→ǫ V (2)
N−1 0
We do not repeat the definition and we refer to [6]§3, for we shall actually
use another chain complex.
3
For q = 0,...,N −1, we consider the G-module
C (X,C) = V
q σ
M
dimσ=q
where the action of g ∈ G is given by g.(vσ)σ = (gvg−1σ)σ. We have boundary
maps
∂ : C (X,C) → C (X,C) , (v ) 7→ (w ) ,
q+1 q τ τ σ σ
where
w = [τ : σ]v .
σ τ
X
τ⊃σ , dimτ=q+1
These boundary maps are Go-equivariant. We have an augmented complex
of chains in the category of Go-modules:
0 → C (X,C)→∂ ···→∂ C (X,C)→ǫ V (3)
N−1 0
where ǫ[(v ) ] = v .
σ σ σ
X
dimσ=0
This complex is isomorphic to the augmented complex of oriented chains
(2) when we see the latter as a complex of Go-modules.
Proposition 3.1 The chain complex (2) is a chain complex in the category
R (G).
(I,1)
Proof. Usingthelabellingλ,anysimplexσ inX givesrisetotwooriented
simplices (cf. [6]§3) hσi and hσ¯i (where hσi = hσ¯i if σ is a vertex). For all
q ∈ {0,...,N −1} and for all q-simplex σ, we write Cor(X,σ) for the space
q
of oriented q-chains with support in {hσi ∪ hσ¯i}. Then we have a natural
isomorphism of P -modules Cor(X,σ) ≃ V .
σ q σ
Fix q ∈ {0,...,N −1}. We must prove that, as a G-module, Cor(X,C)
q
is generated by a subspace fixed by I. Fix a vertex s of C . Then any q-
o
simplex of G has a G-conjugate σ satisfying s ⊂ σ ⊂ C . We fix a system
o
Σ of representatives of the G-conjugacy classes of q-simplices σ satisfying
s ⊂ σ ⊂ C . We have the decomposition
o
Cor(X,C) = gCor(X,σ) ,
q q
M M
σ∈Σg∈G/Gσ
where G ⊃ P is the stabilizer of σ in G. So we are reduced to proving that
σ σ
for all σ ∈ Σ, the G-module gCor(X,σ) lies in R (G). For this it
q (I,1)
M
g∈G/Gσ
4
suffices to prove that Cor(X,σ) is generated by Cor(X,σ)I as a P -module,
q q σ
in other words that V is generated by VI as a P -module.
σ σ σ
We may see V := VUs as a complex representation of G = P /U ≃
s s s
GL(N,k), where k is the residue field of F. Moreover the image P (resp.
σ
B) of P (resp. I) in P /U is a parabolic subgroup of G (resp. a Borel
σ s s
subgroup). The unipotent radical U of P is the image of U in P /U . In
σ σ σ s s
particular we may see V = (V )Uσ as the Jacquet module of V with respect
σ s s
to U . We have the following key result due to Schneider and Zink:
σ
Lemma 3.2 ([7] Prop. (5.3), page 19.) The G-module V has cuspidal
s
support (T,1), where T is the Levi subgroup of B (a maximal split torus in
G).
It follows that V = VU has cuspidal support (T,1) as well. By Frobenius
σ s
reciprocity for parabolic induction, we have that V is generated by VB =
σ σ
(V )I as a P(= P /U )-module and we are done.
σ σ σ
4 The strategy
We know that (2) is a chain complex in R (G). Let H(I,1) denote the
(I,1)
Hecke algebra of (I,1) and Mod(H(I,1)) the category of left unital H(I,1)-
modules. We have an equivalence of categories (cf. [2], [3]):
R (G) −Φ→ Mod(H(I,1))
(I,1)
W 7→ WI .
Soinordertoprovethattheaugmentedchaincomplex(2)isacyclicitsuffices
to prove that the corresponding augmented chain complex in Mod(H(I,1))
is acyclic. We must prove that
0 → Cor (X,C)I −Φ(→∂) ··· −Φ(→∂) Cor(X,C)I −Φ→(ǫ) VI
N−1 0
is exact. But since I ⊂ Go and the complexes (2) and (3) are isomorphic as
complexes of Go-modules, we are reduced to proving:
Proposition 4.1 The following sequence of C-vector spaces is exact:
0 → C (X,C)I −Φ(→∂) ··· −Φ(→∂) C (X,C)I −Φ→(ǫ) VI . (4)
N−1 0
The proof will proceed in several steps.
5
5 Some geometric lemmas
We fix a simplex σ ⊂ C and an element w ∈ Waff. Let E[σ,w−1C ] be the
o o
enclos of σ∪w−1C (cf. [4], Definition (2.4.1)). This is a convex subset of A
o
as well as a subsimplicial complex.
Lemma 5.1 There exists a unique chamber C = C (σ,w) of A such that
1 1
σ ⊂ C ⊂ E[σ,w−1C ] .
1 o
Proof. The element w does not play any special role here and we may
assume that C = w−1C is any chamber of A. Recall [4] that E[σ,C ] is
2 o 2
a union of closed chambers. This proves the existence of C . By [4], Prop
1
(2.4.5), E[σ,C ] is the intersection of the half-apartments (corresponding to
2
affine roots) containing both σ and C . Assume that C and C′ are two
2 1 1
distinct chambers satisfying the assumption of the lemma. Since the set
E[σ,C ] is closed in the sense of [4], it contains all chambers in a minimal
2
gallery connecting C and C′. We may choose such a gallery so that its
1 1
chambers contain σ. It follows that we may assume that C and C′ are adja-
1 1
cent (and contain σ). Let H be a half-apartment whose hyperplan contains
C ∩C′ ⊃ σ. We may choose H so that H contains C . But then H must
1 1 2
contain the enclos E[σ,C ] ⊃ C ∪C , a contradiction.
2 1 2
We let C(σ,w) denote the chamber wC . By equivariance and using the
1
last lemma, C(σ,w) is the unique chamber of A satisfying
wσ ⊂ C(σ,w) ⊂ E[wσ,C ] .
o
It only depends on the simplex wσ (and on the reference chamber C ).
o
Lemma 5.2 With the previous notation, we may choose points x ∈ |σ|o,
σ
x ∈ |C |o, x ∈ |w−1C |o, in the geometric realizations, such that x belongs
1 1 w o 1
to the geodesic segment [x x ].
σ w
Remark. In the terminology of [6]§2, this means that the simplex C lies
1
between σ and w−1C .
o
Proof. Let x be the isobarycenter of σ and C be the union of the open
σ
segments (xx ) where x runs over |w−1C |o. If x is the isobarycenter of
σ o σ
w−1C , then σ = C = w−1C and the lemma is clear. So we assume that x
o 1 o σ
is not the isobarycenter of w−1C so that C is an open subset of A, contained
o
in the enclos E[σ,w−1C ]. There is an open chamber |D|o, contained in
o
6
E[σ,w−1C ], intersecting C and such that x ∈ D; otherwise we would have
o σ
a covering of C by chambers D , = 1,...,r, such that
i
x 6∈ D ⊃ C
σ i
i=[1,...,r
and this would contradict the fact that x is in the closure of C. But since
σ
x ∈ D, we have σ ⊂ D and by unicity in lemma 4, we have D = C .
σ 1
Lemma 5.3 The subgroup hU ,P ∩Iw−1i generated by U and P ∩Iw−1 is
σ σ σ σ
the Iwahori subgroup I wich fixes C .
1 1
Proof. We have U ⊂ I , since σ ⊂ C . Moreover P ∩ Iw−1 fixes the
σ 1 1 σ
enclos E[σ,w−1C ] pointwise and therefore fixes the chamber C . Hence
o 1
P ∩Iw−1 ⊂ I and we have
σ 1
hU ,P ∩Iw−1i ⊂ I .
σ σ 1
Let I1 (resp. I1) denote the pro-unipotent radical of I (resp. of I). Since
1 1
C lies between σ and w−1C , by [6] Prop. (2.5), we have
1 o
Im[I1 ∩P → P /U ] ⊂ Im[(I1)w−1 ∩P → P /U ]
1 σ σ σ σ σ σ
We therefore have the containment:
U [(I1)w−1 ∩P ]U ⊃ U [(I1)∩P ]U .
σ σ σ σ 1 σ σ
Since I = ToI1 and I = ToI1, we get
1 1
U [Iw−1 ∩P ]U ⊃ U [I ∩P ]U = I .
σ σ σ σ 1 σ σ 1
So we have that I is contained in hU ,P ∩Iw−1i, as required.
1 σ σ
We shall need the following lemma.
Lemma 5.4 Let K, J, L be Iwahori subgroups of G whose respective cham-
bers C , C , C lie in the apartment A. Assume that C lies in the enclos
K J L J
E[C ,C ] of C ∪C . Then we have J = (J ∩K)(J ∩L). In particular we
K L K L
have KJL = KL.
Proof. Let Φ (resp. Φaff denote the set of roots (resp. of affine roots) of G
relative to T. We have the standard valuated root datum (T,(Ua)a∈Φ) of G
(cf. [4], §6 and §(10.2)). In particular for all a ∈ Φaff, we have a congruence
subgroup Ua ⊂ Ua, where a is the vector part of a. For any subset Ω of A, we
denote by Φaff(Ω) the set of affine roots which are positive on Ω and minimal
7
for this property, i.e. a ∈ Φaff(Ω) if a(Ω) ⊂ R and a(Ω) − 1 6⊂ R . From
+ +
loc. cit. §6,7, we have that J is generated by To and the U , where a runs
a
over Φaff(C ). More precisely the product map
J
To U −→ J
a
a∈ΦYaff(CJ)
is surjective for any ordering of the affine roots in Φaff(C ). On the other
J
hand since J ∩K (resp. J ∩L) is the fixator of the enclos E[C ,C ] (resp.
J K
E[C ,C ]) in Go, we have that J ∩K (resp. J ∩L) is generated by To and
J L
the U , a running over Φaff(E[C ,C ]) (resp. Φaff(E[C ,C ])). We claim
a J K J L
that any a ∈ Φaff(C ) is positive on C or on C (whence on E[C ,C ] or
J K L J K
on E[C ,C ]); this will give J ⊂ (J ∩K)(J ∩L), as required. Assume that
J L
there exists a ∈ Φaff(C ) such that a is negative on C and C . Then there
J K L
would exist a half-apartment with hyperplan ker(a) which contains C and
K
C but not C . But this would contradict the fact that C lies in the enclos
L J J
of C ∪C .
K L
6 A lemma on idempotents
Let H(G) be the convolution algebra of locally constant complex functions
on G with compact support, that is the Hecke algebra of G. We shall denote
by ⋆ the convolution product. For any Iwahori subgroup K of G, we let e
K
denote the idempotent of H(G) attached to the trivial character of K.
Lemma 6.1 Let W be a smooth representation of G and J be an Iwahori
subgroup of G whose chamber C lies in A. Then the map WJ → WI,
J
v 7→ e .v is an isomorphism of vector spaces.
I
Lemma 6.2 With the notation as above, let K be an Iwahori subgroup of G
whose chamber C lies in the enclos E[C ,C ]. Then the map WJ → WI,
K J I
v 7→ e .v factors through WJ → WK → WI, where the first map is given by
I
v 7→ e .v and the second is given by w 7→ e .w.
K I
Proof. We claim that e ⋆e ⋆e = e ⋆e . The lemma will follow easily.
I K J I J
The support of e ⋆ e ⋆ e is IKJ wich is IJ = supp(e ⋆ e ) by lemma
I K J I J
7. It suffices to prove that e ⋆e ⋆e (1 ) = e ⋆e (1 ), a straightforward
I K J G I J G
computation.
As a consequence of this lemma, we have that in lemma 8, we may reduce
to the case where the chambers C and C are adjacent.
I J
8
Lemma 6.3 With the notation of lemma 8, assume moreover that the cham-
bers C and C are adjacent. Then there exist functions f,g ∈ H(G) such
I J
that:
e = f ⋆e ⋆e and e = e ⋆e ⋆g .
J I J I I J
Let us first show how this latter lemma implies lemma 8 in the case
where C and C are adjacent. Let v ∈ WJ. If e .v = 0, we have f ⋆ e ⋆
I J I I
e .v = e .v = 0, whence v = 0 and f in injective. Let w ∈ WI. We have
J J
w = e .w = e (e ⋆g.w), with e ⋆g.w ∈ WJ and e is therefore surjective.
I I J J I
Proof of lemma 10. (Compare [5] Theorem (2.5)). If N = 1, the result
is trivial and we assume N ≥ 2. We give the proof of the existence of f;
the proof of the existence of g being similar. Set τ = C ∩ C . This is a
I J
simplex of codimension 1. We have I,J ⊂ P so that we may see e and
τ I
e as idempotents in the group algebra of P /U . We fix an isomorphism
J τ τ
P /U ≃ GL(2,k)×T,whereT ≃ (k×)N−2,sothattheimageofI (resp. J)in
τ τ
P /U is B+×T (resp. B−×T), where B+ (resp. B−) is the Borel subgroup
τ τ
of upper (resp. lower) triangular matrices. So we identify e ≃ e ,
I B+×T
eJ ≃ eB−×T. Write U+ and U− for the unipotent radicals of B+ and B−
respectively. We have
eB+×T ⋆eB−×T ⋆eB+×T = 1/|B− ×T| eB+×T ⋆xeB+×T
x∈XB−×T
= 1/q(q−1)N e ⋆ue
B+×T B+×T
uX∈U−
= 1/q(q−1)N(e + e ⋆ue ) .
B+×T B+×T B+×T
u∈UX−, u6=1
Here q = |k| is the size of the residue field. Any u ∈ U−\{1} writes u tsu′ ,
+ +
where u ,u′ ∈ U+, t belongs to the diagonal torus of GL(2,k) and s is the
+ +
matrix
0 1
s = .
(cid:18)−1 0(cid:19)
So we have
eB+×T ⋆eB−×T ⋆eB+×T = 1/q(q−1)N(eB+×T + eB+×T ⋆seB+×T) .
u∈UX−, u6=1
But since eB+×Ts = seB−×T, we get
eB+×T ⋆eB−×T ⋆eB+×T = 1/q(q −1)N(eB+×T +(q −1)seB−×T ⋆eB+×T) ,
9
that is
(q(q−1)NeB+×T −(q −1)s1GL(2,k)×T)⋆eB−×T ⋆eB+×T = eB+×T .
In terms of functions on G, we have proved:
(q(q −1)Ne −(q−1)e )⋆e ⋆e = e
I UτsUτ J I I
and we are done.
7 Computation of the I-fixed vectors
Fix q ∈ {0,...,N −1}. A set of representatives of the Go-conjugacy classes
of q-simplices in X is given by
Σ := {σ simplex of X; σ ⊂ C , dimσ = q} .
q o o o o
If a simplex σ is Go-conjugate to σ ∈ Σ , we say that σ has type σ . The set
o q o
of simplices of types σ is isomorphic to Go/P as a Go-set. Any simplex of
o σo
type σ has a unique I-conjugate lying in the apartment A, so that we have
o
the identification:
I\G/P ≃ {σ ⊂ A ; σ has type σ } .
σo o
On the other hand, Waff acts transitively on the simplices of types σ in A
o
so that we have:
Waff/Waff ≃ {σ ⊂ A ; σ has type σ } ,
σo o
where Waff = {w ∈ Waff ; wσ = σ }. Of course these identifications are
σo o o
compatible with the canonical bijection
Waff/Waff → I\G/P
σo σo
w mod Waff 7→ IwP .
σo σo
For each σ ∈ Σ , we let R denote a set of representatives of Waff/Waff.
o q σo σo
We see Waff as a subgroup of No in the usual way, so that R ⊂ No.
σo
We have the decomposition
C (X,C) = V = V
q σ gσ
diMmσ=q σM∈Σqg∈MGo/Pσ
10