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Coefficient systems and Jacquet modules

Let F be a locally compact non-archimedean field and G the group of F-rational points of an algebraic group assumed to be defined over F, semisimple, simply connected and of F-rank 1. Let pi be a complex irreducible supercuspidal representation of G. We prove that pi is"nearly"induced in the following sense. There exist a maximal compact subgroup K of G and an irreducible smooth representation lamba of K such that pi contains lambda by restriction to K and such that the representation compactly induced from lambda to G is a finite direct sum of irreducible supercuspidal representations. The proof relies on the Schneider and Stuhler theory of equivariant coefficient systems and on a lemma on coefficient systems and Jacquet modules.


Introduction
Let F be a non-archimedean locally compact field of residue characteristic p and G denote the group of F -rational points of a reductive group G defined over F . We assume that G is semisimple, simply connected and of F -rank 1. Let T be the group of rational points of a maximal split torus in G and L be its centralizer. Let P be a minimal parabolic subgroup of G with Levi decomposition P = LU. We denote by A the apartment of the Bruhat-Tits building X of G attached to the torus T . Let C be a field of characteristic l = p. Let F be a G-equivariant coefficient system of C-vector spaces on X in the sense of Schneider and Stuhler [SS].
The homology module H 0 (X, F ) is a smooth representation of G. Our first result is the following (see Theorem (4.3)): there exists an L-equivariant coefficient system G on A, naturally attached to F , such that the Jacquet module H 0 (X, F ) U is isomorphic to H 0 (A, G) as an L-module. The section spaces of G are defined as follows. For any simplex σ of T , the space F σ is a G σ -module in a natural way, where G σ is the stabilizer of σ in G; we then define G σ as the space of G σ ∩ U-fixed vectors in F σ .
This result may be viewed as a generalization of a result of Bushnell and Kutzko which in certain cases gives the Jacquet module of a compactly induced representation as a compactly induced representation ( [BK] Lemma (10.3)).
As an application, we prove the following result.
Theorem 0.1 . Let π be a complex irreducible supercuspidal representation of G. Then there exist a maximal compact subgroup of G, as well as an irreducible representation λ of K, such that: (i) the compactly induced representation c-ind G K λ decomposes as a finite sum π 1 ⊕ π 2 ⊕ · · · ⊕ π s of irreducible supercuspidal representations of G, (ii) π contains the pair (K, λ) by restriction to K, i.e. π is isomorphic to π i for some i = 1, ..., s.
The proof of this result relies on two ingredients. First we use the fact due to Schneider and Stuhler (cf. [SS]) that if G is the group of F -rational points of a reductive group defined over F , then any irreducible smooth representation of G is isomorphic to H 0 (X, F ), for some G-equivariant coefficient system on the building X of G. The second ingredient (Theorem 5.1.2) is the fact that if λ is a finite dimensional representation of a maximal compact subgroup K of G such that λ U ∩K = 0, then the induced representation c-ind G K λ is a finite direct sum of supercuspidal representations of G. The idea of the proof is then the following. We start with an irreducible supercuspidal representation π of G that we write H 0 (X, F ) for some Gequivariant coefficient system F on X. By our first result, we have π U = 0 = H 0 (X, F ) U ≃ H 0 (A, G). As a consequence the boundary map in the chain complex computing H 0 (X, G) is surjective. From this we prove a technical lemma asserting that, in the chain complex ∂ : C 1 (X, F ) −→ C 0 (X, F ) computing H 0 (X, F ), certain special elements of C 0 (X, F ) actually lies in the image of the boundary map ∂. Now, by definition π is a quotient of C 0 (X, F ), which may be written as a sum of certain compactly induced representations.
Certain of these induced representations are good in the sense that they are direct sums of a finite number of supercuspidal representations, the other which do not have this property are the bad ones. Our technical lemma allows then to prove that the bad parts of those compactly induced representations are killed in the quotient. From this it is not difficult to conclude that our Theorem holds.
It is a folklore conjecture that any irreducible supercuspidal representation of a p-adic reductive group should have a type in the sense of [BK] §4. The following result taken from [BK] gives a characterization of types for supercuspidal representations.
Proposition 0.2 ( [BK], Proposition (5.2), page 602.) Let G be the group of F -rational points of a connected reductive group G defined over F , and let π be an irreducible supercuspidal complex smooth representation of G. Let Z denote the center of G, Z the group of its F -rational points and • Z the unique maximal compact subgroup of Z. Finally let K be a compact open subgroup of G containing • Z and ρ be an irreducible smooth representation of K such that the restriction of ρ to • Z is a multiple of the central character of π.
Letρ denote some extension of ρ to ZK. Then (K, ρ) is a type for π if and only if there exist unramified quasicharacters χ 1 , χ 2 , ..., It follows that in our Theorem 0.1, the pair (K, ρ) is a type for π if and only if the representation π i , i = 1, ..., s, are unramified twists of a single representation. Our result, in our framework of semisimple, simply connected, of F -rank 1 groups, may be seen as a first step in proving that irreducible supercuspidal representations have types.
In the litterature, when it is known that a given supercuspidal representation has a type, a stronger result is actually proved: the representation is compactly induced (i.e. s = 1 in our Theorem 0.1). If our result is less precise it avoids using the deep arithmetic structure of the reductive group G and give a uniform proof for the class of groups that we consider. Moreover our result seems to be genuinely new for the group SL(2, D), where D is a central division F -algebra of degree d > 1, and for some non-split forms of Sp 4 , Sp 6 , Spin 6 , Spin 8 and Spin 10 , which are in the isogeny class of special unitary groups over quaternion algebras with involution. This paper is organized as follows. Section 1 consists of some notation and of basic facts on the action of a rank 1 reductive group on its Bruhat-Tits building. In section 2 we review the notion of an equivariant coefficient system on X or A and define the L-equivariant coefficient system G. In section 3 we recall Bushnell and Kutzko's result on the Jacquet module of a compactly induced representation and prove a slight generalization. Section 4 is devoted to the proof of a commutative diagram (Theorem 4.1) that is then applied to the proof of our first result (Theorem 4.3). The proof of Theorem (0.1) is given is section 5. In §5.1 we review the structure of compactly induced representations and we state the general theorem of Schneider and Stuhler. Our technical lemma is proved in §5.2 and the end of the proof of Theorem (0.1) is given in §5.3.
I would like to thank François Courtès and Peter Schneider for their comments, and Guy Henniart for his encouragment to tackle the rank 1 case (a first version of this work was restricted to the case of SL(2, F )). Finally I thank Maarten Solleveld whose remarks helped me to improve a former version of this work.
1 Reductive groups of relative rank 1 For the relative theory of reductive algebraic groups we refer to Chapter V of [Bo]. Basic facts on the Bruhat-Tits building of a p-adic reductive group may be found in [Ti] and [BT]. The aim of this section is to fix the notation and gather together a few lemmas on the action of a rank 1 reductive group on its Bruhat-Tits building that we shall need later in the paper.
We fix an non archimedean locally compact field F . We fix a reductive F -algebraic group G that we suppose semisimple, simply connected and of F -rank 1. If H is an algebraic group defined over F , we shall denote by H the group of its F -rational points.
We fix a maximal F -split torus T in G so that T is isomorphic to the multipicative group G m over F . We denote by L = Z G (T) the centralizer of T in G and N = N G (T) its normalizer. The spherical Weyl group W = N/L is a group with 2 elements. The group L is the Levi component of a minimal parabolic subgroup P defined over F and one has P = L.U (semidirect product), where U = R u (P) is the unipotent radical of P.
The locally compact group L = L(F ) is compact mod center and we denote by L 0 its unique maximal compact subgroup.
We denote by X the Bruhat-Tits building of G in the sense of [Ti]. It is a simplicial complex of dimension 1 (indeed a tree) on which G acts by simplicial automorphisms. For i = 0, 1, we denote by X i the set of idimensional simplices of X. Let A be the apartment attached to T. As a simplicial complex it is isomorphic to the line Z s whose vertex set is the set of integers Z and edge set is We fix such an isomorphism in such a way that any u ∈ U fixes the vertex number k for k large enough. For i ∈ Z, we denote by s i the vertex corresponding to i, and by a i the edge corresponding to {i, i + 1}.
If σ is a simplex of X, we denote by G σ the G-stabilizer of σ. Moreover we somtimes abbreviate G 0 = G s 0 , G 1 = G s 1 and I = G a 0 . Since G is semisimple and simply connected, we have the following facts: -I = G 0 ∩ G 1 is an Iwahori subgroup of G, -the G i , i = 0, 1 are maximal compact subgroups of G, -the action of G on the vertices of X has two orbits: G.s 0 and G.s 1 . The apartment A is the Coxeter complex of an affine Weyl group of rank 1. If, for i = 0, 1, we denote by r i the reflection relative to the wall {s i }, the affine Weyl group of A is the dihedral group generated by r 0 and r 1 . Recall that N acts on A via affine isomorphisms and that the image of the corresponding morphism N −→ Aff(A) is the affine Weyl group of A. For i = 0, 1, we fix an element w i ∈ N which induced the reflection r i on A. The element w 1 w 0 induces the translation t = r 1 r 0 . Let us notice that s k = t k/2 s 0 , if k is an even integer, and s k = t (k−1)/2 s 1 , if k is odd.
Proof. Since A is a Coxeter complex of rank 1, all its vertices are automatically special. In particular we have G = G 0 P = G 1 P . The compact group L 0 ⊂ N fixes A pointwise, so that L 0 ⊂ G i ∩ L, for i = 0, 1. On the other hand, for i = 0, 1, the intersection G i ∩ L is compact. So by maximality of L 0 , we have the equality G i ∩ L = L 0 . Lemma 1.2 Let k ∈ Z. We have the equalities: Proof. The map f : P −→ L, defined by: f (p) is the unique element of L such that there exists u ∈ U satisfying p = f (p)u is well defined and continuous. In particular f (P ∩ G s k )) is a compact subset of L. Since L normalizes U, this is a subgroup of L. On the other hand, we certainly have L 0 ⊂ f (P ∩ G s k , whence by maximality of L 0 , we have f (P ∩ G a 0 ) = L 0 . The lemma follows easily. Lemma 1.3 Let k be a fixed integer. ( (ii) The group G k ∩ U acts transitively on the set of edges a of X such that a contains s k and a = a k .
Proof. (i) We clearly have G a k ∩U ⊂ G s k ∩U. On the other hand if u ∈ G k ∩U, then u fixes s l for l large enough, whence fixes the segment [s k , s l ] pointwise for l large enough. In particular u fixes a k = [s k , s k+1 ], that is u ∈ G a k .
(ii) By [BT] Corollaire (2.2.6), page 36, the stabilizer G a k of a k acts transitively on the set of apartments containing a k . So it acts transitively on the set of edges a containing s k and different from a k . Morover by [BT] Proposition (5.2.11), page 100, we have the Iwahori decomposition G a k = U a k L 0Ū a k , whereŪ is the unipotent radical of the parabolic subgroup opposed to P relative to T andŪ a k = G a k ∩Ū . It follows that Proof. a) This is a classical property of double Tits system: it is given by [BT] Théorème (5.1.3)(vi). b) Moreover since L acts on A via translations, we have w 0 ∈ N\L, so that N/L ≃ Z/2Z is generated by the image of w 0 in N/L. This proves that G = ILU ∪ ILw 0 U and that Ψ is surjective.
For the injectivity of Ψ, we have to prove that for l 1 , l 2 ∈ L, i ∈ I, u ∈ U, we have: ∈Ū the unipotent radical of the parabolic subgroupP opposed to P with respect to the torus T. So (ii) follows from (i) by replacing P byP .
To prove (iii), assume for a contradiction that il 1 = l 2 w 0 u. There exists k 0 ∈ Z, such that u fixes s k for k ≥ k 0 . Then for k ≥ k 0 , we have il 1 s k = l 2 w 0 s k . Two points of A conjugate by I must be equal, so that Lemma 1.5 The set of simplices of the apartment A is a fondamental domain for the action of U on the simplices of X.
Proof. We must prove that for any simplex σ of X there exist u ∈ U and a unique σ 0 ⊂ A such that σ = uσ 0 . If σ is a vertex (resp. an edge), the existence of σ 0 is a consequence of the decompositions G = ULG s 0 , G = ULG s 1 (resp. of the decomposition G = ULNG a 0 ) and of the fact that any vertex of X is conjugate to s 0 or to s 1 (resp. that any edge is conjugate to a 0 ).
For the uniqueness, assume that σ 0 = uσ 0 for some simplex σ 0 of A, and some u ∈ U. Since σ 0 and uσ 0 are two conjugate simplices of A, there exists l ∈ L such that σ 0 = luσ 0 . So lu ∈ G σ 0 . If σ is a vertex, we apply Lemma 1.2 to obtain u ∈ G σ 0 , so that uσ 0 = σ 0 as required. If σ 0 = {s, t} is an edge, we have lu ∈ G s ∩ G t and we conclude by applying Lemma 1.2 twice.

Equivariant coefficient systems
We fix a commutative field C. We denote by S(G) the category of smooth representations of G in C-vector spaces. In this section we make no assumption on the characteristic of C and we do not assume that it is algebraically closed.

G-equivariant coefficient systems on X
Following Schneider and Stuhler [SS], we define a coefficient system F (of C-vector spaces) on X to be a collection ((F σ ) σ , (r σ τ ) τ ⊂σ ), where: -for each simplex σ of X, F σ is a C-vector space, -for each pair of simplices τ ⊂ σ, r σ τ : -for all g, h ∈ G, any σ simplex of X, we have -for any g ∈ G, for any pair of simplices τ ⊂ σ of X, we have If F is a G-equivariant coefficient system on X, for any simplex σ, we shall denote by ρ σ the natural representation of G σ in F σ .
We fix incidence coefficients on X in the following way. If a is an edge and s a vertex, we set [a : s] = 0, if s ∈ a, [a : s] = 1 is s ∈ a and s ∈ G.s 0 , and [a : s] = −1 if s ∈ a and s ∈ G.s 1 . Since G is simply connected, its action on X preserves incidence coefficients : such that f has finite support and for any i-dimensional simplex σ, f (σ) ∈ F σ . We have a boundary operator ∂ : The chain complex is by definition the chain complex of X with coefficients in F . It is a complex in the category S(G) and its homology spaces H i (X, F ) ∈ S(G), i = 0, 1, are by definition the homology spaces of X with coefficients in F . They are smooth representations of G.
Remark. The notion of an equivariant coefficient complex extends naturally to the case where X is any simplicial complex endowed with an action of a locally profinite group G via simplicial automorphisms. If moreover X possesses an orientation with G-invariant incidence numbers, we may in the same way define the chain complex of X with coefficients in any G-equivariant coefficient system F , as well as the homology groups H * (X, F ). In particular we have the notion of a T -equivariant coefficient system on A .
We now describe the chain complex in terms of induced representations. Let us be given the compactly induced representation c where G acts on functions by right translation.
Similarly, one may consider the two induced representations c-ind G given by is an isomorphism of G-modules, whose inverse is given by Proof. Straightforward.

L-equivariant coefficient systems on
For any simplex σ of A, we denote by ρ L σ the natural representation of the stabilizer L σ of σ in L (indeed L σ = L 0 ). Contrary to the case of X, which admits a single G-orbit of edges, A has two L-orbits of edges: that of a 0 and that of a −1 = {s −1 , s 0 }.
The following lemmas whose proofs are straightforward will be useful.
is an isomorphism of L-modules whose inverse is given by We shall attach to F an L-equivariant coefficient system on A in a natural way.
For any simplex σ of X, we set U σ = U ∩ G σ . We define a collection G = ((G σ ) σ , (R σ τ ) τ ⊂σ , (ψ l,σ )) in the following way: -for any simplex σ of A, we set G σ = F Uσ σ , the set of vectors fixed by -for any pair of simplices τ ⊂ σ in A, we define a linear map R σ τ : -for l ∈ L and σ a simplex of A, we define ψ l,σ : G σ −→ G lσ as the linear map induced by ϕ l,σ .
is well defined and is an L-equivariant coefficient system on A.
Proof. Easy and follows mainly from the equality U lσ = lU σ l −1 due to the fact that L normalizes U.

Compact induction and Jacquet modules
The aim of this section is to describe the Jacquet module of a compactly induced representation as a (sum of) compactly induced representation(s). This was done by Bushnell and Kutzko in [BK] (Lemma 10.3 of §10, page 628). We shall in fact need a slightly more general version of their lemma. In this section we assume that the characteritics l of the field C is different from the residue characteristics p of F .
In this section only, G denotes the group of F -rational points of a connected reductive F -algebraic group. We fix a parabolic subgroup P with Levi decomposition P = MU, as well as a compact open subgroup K of G. We make the following assumption: Lemma 3.1 We have the equalities: Proof If (π, V) is a smooth representation of G in a C-vector space V, we define its Jacquet module (π U , V U ) as the natural Since U is a pro-p-group, we may and do fix a C-valued Haar measure µ on U. Now, with the notation of loc. cit., take a smooth representation (ρ, W ) of K. Let ρ M be the representation of K M = (K ∩ P )/(K ∩ U) on the space W K∩U of ρ(K ∩ U)-fixed vectors in W .
Let us realize the representaton c-ind G K ρ in the usual model V of certain functions on G with values in W . Following Bushnell and Kuztko, for f ∈ V, we define Φf : M −→ W by is surjective. Morever, under the assumption that G = KP , it is injective.
Proof. The bijectivity of Φ when G = KP is [BK], Lemma 10.3. We just have to observe that Bushnell and Kutzko do not use the assumption G = KP to prove the surjectivity of Φ.
Remarks 1. In their Lemma 10.3 of [BK], Bushnell and Kutzko assume that W is finite dimensional. However they never use this assumption is their proof.
2. Bushnell and Kutzko normalizes µ in such a way that µ(K ∩ U) = 1. The effect of another normalization multiplies Φ by a non-zero constant.
We now specialize to the case of our F -rank 1 group G. Here we take P = LU, the minimal parabolic subgroup introduced in §1. We consider the two compact open subgroups I = G a 0 and w −1 0 I = w −1 0 Iw 0 = G a −1 . These Iwahori subgroups are not special, but they do satisfy assumption (A). For i = 0, −1, we set U a i = U ∩ G a i . We fix a smooth representation (ρ, W ) of I and write ρ w −1 0 for the representation of w −1 0 I given by ρ w −1 0 (w 0 −1 kw 0 ) = ρ(k). The representations c-ind G I ρ and c-ind G w −1 0 I ρ w −1 0 are isomorphic and a specific intertwining operator is f → f w 0 , where f w 0 is defined by f w 0 (g) = f (w 0 g).
With the notation as above, we have K M = L 0 , if K = I or w −1 0 I. Let ρ L denote the natural representation of L 0 in the space of ρ(U a 0 )-fixed vectors in ρ, and similarly let ρ Finally we consider the following map: Proposition 3.3 . The map Φ induces a bijective isomorphism of L-modules: We closely follow the proof of [BK] Lemma (10.3), by replacing the assumption "G = KP ", which does not hold in our case, by Lemma 1.4.
Let us first prove the surjectivity of Φ. Write (π, V) for the induced representation c-ind G I ρ. Then V is generated as a C-vector space by the f g,v , g ∈ G, v ∈ W , where f g,u is the function with support Ig given by f g,v (kg) = ρ(g).v, k ∈ I. Since for u ∈ U and f ∈ V, we have Φ(π(u)f ) = Φ(f ), and using the identity π(u)f g,v = f gu,v , we deduce that the image of Φ is generated as a C-vector space by the functions Φ(f g,v ), where u ∈ U and g runs over a system of representatives of I\G/U. Moreover by the previous lemma, we may make g run over a system of representatives of L 0 \L ∪ (L 0 \L)w 0 . Hence we first calculate the images of functions of the form f l,v , f lw 0 ,v , l ∈ L, v ∈ W .
Write p 0 (resp. p −1 ) for the canonical U a 0 -projection W −→ W ρ(Ua 0 ) (resp for the canonical U a −1 -projection W −→ W ρ w −1 0 (Ua −1 ) ). For l ∈ L, v ∈ L , with support L 0 l, by the formulas: Lemma 3.4 For l ∈ L and v ∈ W we have: Proof. For x ∈ L, we first compute where f = f l,v and δ P denote the modulus character of P . For u ∈ U, if f (ux) = 0 then u ∈ Ilx −1 , whence 1 ∈ Ilx −1 U. By lemma 1.4 this implies lx −1 ∈ L 0 . Therefore the support of Φ 0 f lies in L 0 l = lL 0 . Write x = l 0 l, l 0 ∈ L 0 . We have: where we used the change of variable u → (l 0 ) −1 ul 0 and the fact that δ P (l 0 ) = 1. Now we have f (ul) = 0 if and only if u ∈ I ∩ U so that as required.
It follows from the lemma that Φ is onto.
. One may write f as a linear combination Moreover changing f by an element of V[U], we may assume that f writes and where the l i (resp. the m j ) are pairewise distinct elements in a set of representatives of L 0 \L. Then so that for all i, F l i ,p 0 (v i ) = 0, and for all j, H w −1 0 m j w 0 ,p −1 (v ′ j ) = 0. So we may as well as assume that f has the form f l,v , or f lw 0 ,v , for some l ∈ L and some v ∈ W .
In the first case, we get F l,p 0 (v) = 0, whence p 0 (v) = 0. It follows that v writes we obtain : as required. The second case is similar. This achieves the proof of the proposition.

Coefficient systems and Jacquet modules
In this section we assume that the characteristic l of C is not p and we fix a C-valued Haar measure µ on U.
We fix a G-equivariant coefficient system F = ((F σ ), (r σ τ ), (ϕ σ,g )) on X. For any simplex σ of X, we set U σ = U ∩ G σ , and we denote by ρ Uσ σ the natural representation of L 0 = L ∩ G σ in the space F Uσ σ of ρ σ (U σ )-invariant vectors in F σ By Lemmas 1.1 and 3.2, we have an isomorphism of L-modules: By Proposition 3.3, we have an isomorphism of L-modules: We denote by Ψ 1 : a −1 . It follows that we have an isomorphism of L-modules: . Let G be the L-equivariant coefficient system on A attached to F as in Lemma 2.2.2. Using Lemma 2.2.1, we have natural L-isomorphisms: Theorem 4.1 Write ∂ A for the boudary map C 1 (A, G) −→ C 0 (A, G). Then the following diagram is commutative: where ∂ U is the map induced by the boundary map ∂ : C 1 (X, F ) −→ C 0 (X, F ) on Jacquet modules.
We give a proof for i = 0, the other case being similar.
For any ω ′ ∈ C 1 (A, F ) and l ∈ L, we have : Therefore we have by the change of variable v = l −1 ul. On the other hand, for s ∈ X 0 , we have and ϕ 0 (∂(ω))(ls 0 ) = δ P (l) by the change of variable v = l −1 ul. By Lemma (1.2)(ii), the edges of X containing s 0 are a 0 and the ua −1 , where u runs over a set of representatives of G s 0 ∩U/G a −1 ∩U = U s 0 /U a −1 . Moreover by Lemma (1.2)(i), we have r a 0 s 0 = R a 0 s 0 . So we obtain where the term Σ is given by by the change of variable w = u −1 v. By definition of the restriction map R a −1 s 0 , we finally obtain and this proves the commutativity of the diagram.
Theorem 4.3 Let F be a G-equivariant coefficient system on X. Let G be the L-equivariant coefficient system on A attached to F as in §2.2. Then we have an isomorphism of L-modules: Proof. This is a direct consequence of Theorem 4.1 and of the exactness of the Jacquet module functor.

Application to supercuspidal representations
In this section the field C is assumed to be algebraically closed and of characteristic 0. We now apply Theorem 4.3 to supercuspidal representations. More precisely, we shall prove the following result.
Theorem 5.1 Let π be an irreducible supercuspidal representation of G. Then there exists a maximal compact subgroup K ∈ {G s 0 , G s 1 }, an irreducible smooth representation λ of K such that: i) c-ind G K λ is a finite sum π 1 ⊕ π 2 ⊕ · · · ⊕ π s of irreducible supercuspidal representations.

Some useful general results
The proof of Theorem (5.1) relies on two results that we review in this section. These results are quite general and hold for any p-adic reductive groups.
We first need to review the Bernstein decomposition of the category S(G) of smooth representations of G. This decomposition is actually available for any connected reductive group over F . For more details the reader may refer to [BK] §1, or to [Be] §2. Recall that a cuspidal pair in G is a couple (M, σ) formed of a Levi subgroup M of G and of an irreducible supercuspidal representation σ of M. An unramified character of M is a character of the form g → |χ(g)| s F , where χ : M −→ G m is an F -rational character of M, s a complex number, and where | − | F denotes the absolute value of F (normalized in such a way that e.g. |̟ F | F = 1, for any uniformizer ̟ F of F ). Let us notice that any unramified character of G is trivial. Two cuspidal pairs (M i , σ i ), i = 1, 2, are called inertially equivalent if there exist g ∈ G and an unramified character φ of L 2 such that M 2 = gM 1 g −1 and σ 2 ≃ σ g 1 ⊗ φ. We let B(G) denote the set of equivalence classes [M, σ] of cuspidal pairs. For s = [M, σ] ∈ B(G), we denote by S s (G) the full subcategory of S(G) whose objects are the smooth representations π of G satisfying: any irreducible subquotient π ′ of π is a subquotient of a parabolically induced representation ind G P σ ⊗ φ, for some parabolic subgroup P of G with Levi component M, and some unramified character φ of M.
Then the category S(G) decomposes as the direct sum: We shall need the following finiteness result.
Proposition 5.1.1 Let K be an open compact subgroup of G. Let (π, V) be an object of S(G) such that V is generated by V K as a G-module. Let be the decomposition of π with respect to (2). Then for all but a finite number of s, we have V s = 0.
Theorem 5.1.2 Let K = G s i for i = 0 or 1. Let (λ, W ) be a finite dimension smooth representation of K. Assume that W has no non-zero fixed vectors under ρ(U ∩ K). Then (π, V) = c-ind G K λ is a finite sum of irreducible supercuspidal representations.
Proof. By the Bernstein decomposition of S(G), we may write V = V cusp ⊕ V ∞ , where V cusp is supercuspidal, and where no irreducible subquotient of V ∞ is supercuspidal. By Lemma (3.2) we have π U = 0. Hence the representation V is supercuspidal so that V ∞ = 0. If K is a good open compact subgroup of G contained in the kernel of λ, we have that V is generated by V K .
By applying Proposition 5.1.1, there exists a finite number σ 1 , ..., σ s of supercuspidal representations of G such that π = i=1,...,s π i with π i ∈ S [G,σ i ] , i = 1, ..., s. Since π is finitely generated, so is each π i . It follows that the representations π i are finitely generated and cuspidal, whence admissible by [BZ] Corollay (2.41). Since any irreducible subquotient of π i is isomorphic to σ i , it follows that π i is of finite length. The connected center of G being trivial, any supercuspidal irreducible representation of G is a projective object of S(G) ( [Cas] Theorem 5.4.1). Now a basic inductive argument shows that each π i is a direct sum of a finite number of irreducible representations isomorphic to σ i . The theorem follows.
Theorem 5.1.3 (Schneider and Stuhler) Let G be the group of F -rational points of a reductive group defined over F . Let X be it semisimple Bruhat-Tits building. Then for all irreducible smooth representation (π, V) of G, there exists an equivariant coefficient system F on X such that π ≃ H 0 (X, F ) and such that for all polysimplex σ of X, the space of sections F σ is finite dimensional.
Proof. The existence of a G-equivariant coefficient system F such that π ≃ H 0 (X, F ) is given by Theorem II.3.1 of [SS]. In fact Schneider and Stuhler do not construct a single coefficient system F but a family depending on an integer e choosen large enough.
Moreover for any polysimplex σ, the space F σ is the set of vectors in V fixed by a certain congruence subgroups of the parahoric subgroup fixing σ. Since this congruence subgroup is open and since π is admissible, F σ is indeed finite dimensional.
Note that the coefficient system F of the theorem is far from being unique.

A technical lemma
For i ∈ {0, 1} and v ∈ F Us i , let α = α i,v be the 0-chain of F with support {s i } and defined by α(v) = v.
Proof. Since H 0 (X, F ) U = 0, we have H 0 (A, G) = 0 by the isomorphism of Theorem 4.3. It follows that the boundary map ∂ A : C 1 (A, G) −→ C 0 (A, G) is surjective. We may view α as a 0-chain of G, and there exists ω T ∈ C 1 (A, G) such that α = ∂ T (ω T ).
The support Supp (ω T ) of ω T consists of a finite number of vertices. Hence there exist two integers k < l such that Supp (ω T ) ⊂ [s k , s l ] := {s k , s k+1 , ..., s l−1 , s l } .
We are going to define a 1-chain ω ∈ C 1 (X, F ). For this we have to give a value to ω(a) for all edges a ∈ X 1 .
Case 1. Assume first that a ∈ A. We set ω(a) = 0 is a does not lie on the geodesic segments [s k , s l ]. If a = [s i−1 , s i ], i ∈ {k + 1, k + 2, ..., l}, we set ω(a) = ω T (a). Case 2. Assume that a ∈ A. Let m be the middle of the edge a, and s j , for some j ∈ Z, be the projection of m on the apartment A (i.e. the unique vertex s of A which makes the distance d(m, s) minimal). If j > l, we set ω(a) = 0. If j ≤ l, there exist u ∈ U and a unique i ≤ l such that a = u.[s i−1 , s i ]. Here we used the fact that the apartement is a fundamental domain for the action of U on the simplices of X (Lemma 1.5). We then set ω(a) = ϕ a T ,u .ω(a T ), where a T = [s i−1 , s i ], and ω(a T ) is defined as in case 1.
The chain ω is well defined. Indeed, with the notation of Case 2, assume that we have a = u 1 .a T = u 2 .a T , then u −1 2 u 1 ∈ G a T ∩ U = U a T . So ϕ a T ,u 1 .ω(a T ) = ϕ a T ,u 2 u −1 2 u 1 .ω(a T ) = ϕ a T ,u 2 • ϕ u −1 2 u 1 ,a T .ω(a T ) = ϕ a T ,u 2 .ω(a T ) since ω(a T ) is fixed by the action of U a T .
Moreover ω has finite support. Indeed if ω(a) = 0, then with the notation of the definition, the projection of m onto A is s m , for some m ≤ l, and a may be written a = u.a T , with u ∈ U and a T belonging to the finite set of edges Moreover any u such that a = u.a T must fix s k and therefore lies in the compact group U s k . From this we deduce that Supp (ω) is finite. It remains to prove that ∂(ω) = α, that is: We split the proof of this equality in three cases.
Case 1. Assume that the projection of s onto A is s i for some i > l. Then α(s) = 0 and by construction ω(a) = 0 for all ω containing s, so that the equality trivially holds.
Case 2. Assume that s = s i , for some i ≤ l. By assumption we have a∈A 1 , a∋s [a : s]R a s ω T (a) = α(s) .
In the set {a ∈ A 1 ; a ∋ s}, there is one edge a + = [s i , s i+1 ] such that U a + = U s , and an edge a − = [s i−1 , s i ] such that U a − = U s . By definition of the restriction maps R, Equality (23) The set {a + , u.a − ; u ∈ U s /U a − } is precisely the set of edges of X containing s. Using the definition of ω, we obtain: a∈X 1 , a∋s [a : s]r a s ω(a) = α(a) as required.
Case 3. Assume that s ∈ A and that the projection of s onto A write s j , for some j ≤ l. Write s = u.s i with i ≤ l, u ∈ U, so that {a ∈ X 1 ; a ∋ s} = u. {b ∈ X 1 ; b ∋ s i } .

Proof of Theorem (5.1)
Let π be an irreducible supercuspidal representation of G. Using Theorem (5. 1.3), fix a G-equivariant coefficient system F on X, with finite dimensional section spaces, such that π ≃ H 0 (X, F ) as G-modules. For i = 0, 1 we shall identify the G s i -module F s i as the G s i -submodule C 0 (X, F ) s i of C 0 (X, F ) formed of those chains whose support is contained in {s i }. A natural isomorphism is given by In this way, we identify C 0 (X, F ) with the direct sum of the two induced representations c-ind G Gs i F s i = g∈G/Gs i g.F s i , i = 0, 1. Moreover for i = 0, 1, we abbreviate G s i = G i and F s i = λ i . For i = 0, 1, we decompose the G i -module λ i as where the subrepresentation λ cusp i has no non-zero fixed vector by U s i , and where the λ j i , j = 1, ..., n j are irreducible subrepresentations satisfying (λ j i ) Us i = 0.
Proof. Pick a non zero vector v ∈ (λ j i ) Us i ⊂ F U i s i . Under our identifications, it corresponds to the 0-chain α i,v with support {s i } and satisfying α i,v (s i ) = v. Since π is supercuspidal, we have H 0 (X, F ) U ≃ π U = 0. Thus we may apply Lemma (5.2.1): this cochain α i,v lies in the image of ∂. Since v generates λ j i as a G s i -module and since ∂ is G s i -equivariant, we deduce that λ j i lies in the image of ∂. Proof. By assumtion we have π ≃ C 0 (X, F )/Im ∂. Moreover the G-module C 0 (X, F ) decomposes as c-ind G G 0 λ cusp 0 ⊕ c-ind G G 1 λ cusp 1 ⊕ i=0,1 j=1,...,n i c-ind G G i λ j i .
For i = 0, 1, j = 1, ..., n i , we have that λ j i ⊂ Im (∂). Since c-ind G G i λ j i is generated by λ j i as a G-module, and since ∂ is G-equivariant, we have c-ind G G i λ j i ⊂ Im (∂). The lemma follows.
By Theorem 5.1.2, for i = 0, 1, we may write where m i is some postive integer and the π j i , j = 1, ..., m i are irreducible supercuspidal representations. So there exist i ∈ {0, 1} and j ∈ {1, ..., m i } such that π is isomorphic to π j i . Fix such an integer i ∈ {0, 1} and decomposes λ cusp i as a direct sum of irreducible G i -modules: By Frobenius reciprocity, there exists k ∈ {1, ..., l i } such that π is a quotient of c-ind G G i σ k i . The irreducible representation (G i , σ k i ) satisfies the assumption of Corollary (5.1.3) so that c-ind G G i σ k i is a finite direct sum of irreducible supercuspidal representations. Of course π is one of those representations. This finishes the proof of Theorem (5.1).