ON THE COHOMOLOGY OF DRINFEL’D’S p-ADIC SYMMETRIC DOMAIN

GIL ALON AND EHUD DE SHALIT

Let K be a finite extension of ℚ_{p}, V a d + 1-dimensional vector space over K, and V^{∗} its dual. Drinfel’d’s p-adic symmetric domain of dimension d, 𝔛, is the complement in ℙ(V^{∗}) of the K-rational hyperplanes, regarded as a rigid analytic space over K. We denote by A = ℙ(V_{K}) the set of K-rational hyperplanes. If it is necessary to distinguish between a ∈ A and the hyperplane in ℙ(V^{∗}) defined by the equation a(x) = 0, we write H_{a} for the latter. Thus

(0.1)  𝔛 = ℙ(V^{∗}) − ⋃_{a∈A} H_{a}.

The group G = P GL(V_{K}) acts on 𝔛 by rigid motions, and therefore acts on its (rigid) de-Rham cohomology H_{dR}^{·}(𝔛/K), which becomes an infinite-dimensional representation of G. The graded algebra H_{dR}^{·}(𝔛/K) has a natural subalgebra, the algebra of logarithmic classes, generated by the classes [d log(a/b)] of logarithmic 1-forms. (Note that a/b, for a, b ∈ A, is a well-defined and nowhere vanishing function on 𝔛.)

In [S-S] Schneider and Stuhler computed H_{dR}^{·}(𝔛/K) as a G-module. A different approach was proposed in [dS], using residues to identify the cohomology with the space C_{har}^{·} of harmonic cochains on the Bruhat-Tits building T of G. It was shown there that the logarithmic classes are dense in H_{dR}^{·}(𝔛/K), and their associated harmonic cochains are given by an explicit combinatorial formula. However, the relationship between the results of [dS] and those of Schneider and Stuhler was not determined up till now.

Recently [I-S], Iovita and Spiess managed to compute the isomorphism of Schneider and Stuhler on the logarithmic classes, and obtained a third description of the cohomology, which stimulated our present work.

Our goal is to relate the three approaches. We first re-prove the Main Theorem of [I-S] by the methods of [dS]. The derivation is simpler than that of [I-S], which relies on the definitions of [S-S], and is logically independent of it. As a corollary we get the desired comparison between the isomorphisms of [S-S] and [dS]. We shall also clarify the description of H_{dR}^{k}(𝔛/K) by means of boundary distributions on a certain component B_{k−1}^{min} of B_{k−1}, the space of flags of length k in V_{K} (which is closely related to the Borel-Serre boundary of T ).

An important proper subspace of the cohomology is the space H_{dR}^{k}(𝔛/K)_{bnd} of bounded classes (called bounded logarithmic in [I-S]). Under the isomorphism with C_{har}^{k} it corresponds to the bounded harmonic cochains, and thus contains all the logarithmic classes. We show that two natural Banach norms on this space - the sup norm on the bounded harmonic cochains, and the Banach quotient norm on the space of measures on A^{k+1} modulo degenerate measures, coincide. As explained in [I-S], a nice feature of H_{dR}^{k}(𝔛/K)_{bnd} is that each bounded cohomology class has a canonical closed form representing it. We call the space of these representatives Ω_{bnd}^{k}(𝔛). One may ask whether H_{dR}^{k}(𝔛/K)_{bnd} is a maximal subspace in the cohomology which is represented G-equivariantly on the level of differential forms. We do not know the answer to this question. The formula which recovers the form representing a bounded cohomology class from the associated boundary measure, may be considered the p-adic Poisson integral. For k = d it was written down, in coordinates, by Schneider and Teitelbaum in [S-T].

Date: November 15, 2000.

A key technical tool is Proposition 5 of [S-S], Section 3, whose proof the authors attribute to Deligne, and which we therefore called the Lemma of Deligne-Schneider-Stuhler. It allows one to interpolate between the cochain complexes of two profinite simplicial sets, one leading to a description of H_{dR}^{k}(𝔛/K) in terms of a quotient of the space of distributions on A^{k+1}, and the other as a subspace of the space of distributions on B_{k−1}^{min}.

In the final section we review the application that Iovita and Spiess give to their Main Theorem, namely to the cohomology of Γ\𝔛, when Γ is a discrete and cocompact subgroup of G. Recall that by a theorem of Mustafin, this is the rigid analytic space associated to a unique smooth and projectve variety X_{Γ} over K. The de-Rham cohomology of X_{Γ} admits, besides the usual Hodge filreation F_{dR}^{·}, another filtration F_{Γ}^{·}, induced by the covering spectral sequence. The theorem of Iovita and Spiess (formerly a conjecture of Schneider) says that these two filtrations are opposite, hence induce a Hodge-like direct sum decomposition of the cohomology. In [dS], Section 9, we showed that the two spectral sequences in question are the two spectral sequences attached to a certain natural double complex. Here we want to explain how the isomorphism Ω_{bnd}^{k}(𝔛) ≃ H_{dR}^{k}(𝔛/K)_{bnd} leads to a short new proof of the degeneration of the two spectral sequences, as well as of Schneider’s conjecture on the two induced filtrations.

1. The various approaches to the cohomology

1.1. de-Rham cohomology. We use the notation and the definitions of [dS]. Let K, G and 𝔛 be as in the introduction. Let T be the Bruhat-Tits building of G, and r : 𝔛 → |T | the reduction map from 𝔛 to the topological realization of T . Since 𝔛 is a Stein domain (in the sense of Kiehl), its rigid de-Rham cohomology H_{dR}^{k}(𝔛/K) (0 ≤ k ≤ d) is the space of closed k-forms modulo the exact ones. We denote by [ω] the cohomology class of a closed form ω. Recall that |T | has the structure of a metric space, in which every apartment is a Euclidean space. Fix a vertex v of T , let B_{n}(v) be the subcomplex of |T | spanned by the vertices at distance at most n from v, and 𝔛_{n}(v) = r^{−1}(int(B_{n}(v))). Each 𝔛_{n}(v) is a Stein space whose cohomology is finite dimensional, and

(1.1)  H_{dR}^{k}(𝔛/K) = lim_{←} H_{dR}^{k}(𝔛_{n}(v)/K).

The inverse limit topology on H_{dR}^{k}(𝔛/K) is then independent of the choice of v, and G acts continuously.

1.2. First description of the cohomology (Schneider-Stuhler). To follow the approach of [S-S] we have to introduce coordinates. Fix the basis (e_{0}, …, e_{d}) of V_{K}, and identify G with G = P GL_{d+1}(K) as in the paragraph preceding theorem 8.3 of [dS]: g ∈ G corresponds to g = (a_{ij}) ∈ G if

(1.2)  g^{−1}e_{i} = ∑_{j=0}^{d} a_{ij}e_{j}.

If we put

L_{0} = ∑_{i=0}^{d} O_{K}e_{i},

(1.3)  W_{i}^{0} = ∑_{j=d+i−k}^{d} Ke_{j},  L_{i} = πL_{0} + (W_{i}^{0} ∩ L_{0}) (1 ≤ i ≤ k)

then the standard k-simplex

(1.4)  σ_{0} = (L_{0} ⊃ L_{1} ⊃ ⋯L_{k} ⊃ πL_{0})

is of minimal type (i.e. dim L_{i}/L_{i+1} = 1 for i ≥ 1; L_{k+1} = πL_{0}). Let I = {1, …, d − k} and write

(1.5)  B_{I} = Stab_{G}(σ_{0})
(1.6)  P_{I} = Stab_{G}(W_{1}^{0} ⊃ ⋯ ⊃ W_{k}^{0})

for the standard parahoric and parabolic subgroups of G “of type I”. Compare [S-S], p.72, and note that our (e_{0}, …, e_{d}) are denoted in that paper (e_{1}^{∗}, …, e_{d+1}^{∗}). According to our conventions V_{K}^{∗} = K^{d+1} is represented by column vectors, while V_{K} = (K^{d+1})^{∗} is represented by row vectors, and the contragredient action of G on V_{K}^{∗} is the usual left action of G on K^{d+1}. Let

(1.7)  C_{SS}^{k} = Hom(C_{c}^{∞}(G/B_{I}, ℤ)/R_{I}, K),

where R_{I} is the G-module of relations introduced in [S-S]. The isomorphism of Schneider and Stuhler ([S-S], corollary 17, p.83) is a certain (non-explicit) isomorphism

(1.8)  c^{SS} : H_{dR}^{k}(𝔛/K) ≃ C_{SS}^{k}.

1.3. Second description of the cohomology (as harmonic cochains on T ). Let C_{har}^{k} be the space of harmonic cochains on T̂_{k}, as defined in [dS], Section 3.1. The main theorem of [dS] asserts that there is an isomorphism

(1.9)  c : H_{dR}^{k}(𝔛/K) ≃ C_{har}^{k}, [ω] ↦ c_{ω}.

It is defined by the theory of residues: c_{ω}(σ) is the residue of ω along the oriented simplex σ. Furthermore, if S = (a_{0}, …, a_{k}) ∈ A^{k+1} and

(1.10)  ω = ω(S) = ∑_{i=0}^{k}(−1)^{i}d log a_{0} ∧ ⋯ ∧ { \atop d log a_{i}} ∧ ⋯ ∧ d log a_{k},

then

(1.11)  c_{ω}(σ) = (σ, S)

where (σ, S) ∈ {0, ±1} is determined according to the following rule. Assume that σ is defined by a lattice-flag as in (1.4) above. Normalize all the a_{i} to lie in L_{0}−πL_{0}, and let ι_{σ}(a_{i}) denote the last j such that a_{i} ∈ L_{j}. Then if the ι_{σ}(a_{i}) are all distinct, (σ, S) is the sign of the permutation i ↦ ι_{σ}(a_{i}), and otherwise it is 0.

Let C_{bnd}^{k} denote the bounded harmonic cochains. In the sup-norm

(1.12)  ||c|| = sup_{σ∈T̂_{k}} |c(σ)|

it becomes a K-Banach space.

1.4. Third description of the cohomology (Iovita-Spiess). We first need some notation concerning measures and distributions. If X is any locally compact, totally disconnected topological space, we denote by L(X) = C_{c}^{∞}(X, ℤ) the space of all ℤ-valued, compactly supported, locally constant functions on X. For any ring (or abelian group) R we call D(X, R) = Hom(L(X), R) the space of R-valued distributions on X. If μ is a distribution and f ∈ L(X) we denote μ(f) by ∫ f(x)dμ(x). We shall be interested specifically in K-valued distributions. A bounded distribution, which by definition is an element of M(X, K) = D(X, O_{K}) ⊗_{O_{K}} K, is called a measure. If μ is a measure, one can integrate against it not only locally constant, but also continuous, compactly supported K-valued functions.

Our third description of the cohomology goes in a direction opposite to the first two. One starts with a K-valued distribution μ on A^{k+1}, where from now on we endow A with the p-adic topology. The degenerate distributions D(A^{k+1}, K)_{deg} are the μ’s annihilating

(1.13)  Λ′_{k} = Ker(ι^{∗}, ∂^{∗}) : L(A^{k+1}) → L((A^{k+1})_{0}) ⊕ L(A^{k+2}).

Here (A^{k+1})_{0} is the closed subset of all (a_{0}, …, a_{k}) which are linearly dependent, and ι^{∗} is the restriction map. The second map, ∂^{∗}, is the usual coboundary homomorphism. The degenerate measures are the measures which are degenerate as distributions.

To μ Iovita and Spiess attach the integral

(1.14)  Φ(μ) = ∫_{A^{k+1}}[ω(S)]dμ(S) ∈ H_{dR}^{k}(𝔛/K).

The point behind this definition is that the restriction to each 𝔛_{n}(v) of the class of ω(S) modulo exact forms is locally constant (in S), so the integral makes sense. Moreover, if μ is a measure, then since S ↦ ω(S) is continuous, and since continuous functions can be integrated against measures (by Riemann integration),

(1.15)  ω(μ) = ∫_{A^{k+1}} ω(S)dμ(S)

is well defined, and Φ(μ) = [ω(μ)].

It is easy to verify that Φ(μ) = 0 if μ is degenerate. The Main Theorem of [I-S] is that it induces an isomorphism

(1.16)  Φ : D(A^{k+1}, K)/D(A^{k+1}, K)_{deg} ≃ H_{dR}^{k}(𝔛/K).

In more detail, let Y_{·}^{(k)} be the profinite simplicial set given by

(1.17)  Y_{r}^{(k)} = {S = (a_{0}, …, a_{r}) ∈ A^{r+1}; dim(Span(S)) ≤ k} ,

and let |Y_{·}^{(k)}| be its topological realization (see [S-S], p.65). As explained in Section 2 (see 2.7), there is a canonical isomorphism

(1.18)  H̃^{k−1}(|Y_{·}^{(k)}|, ℤ) ≃ Λ′_{k},

for the reduced cohomology, and therefore

(1.19)  Hom(H̃^{k−1}(|Y_{·}^{(k)}|, ℤ), K) ≃ D(A^{k+1}, K)/D(A^{k+1}, K)_{deg}.

On the other hand, as Schneider and Stuhler show, there is a canonical isomorphism

(1.20)  C_{SS}^{k} ≃ Hom(H̃^{k−1}(|Y_{·}^{(k)}|, ℤ), K).

Combining (1.20), (1.19) and (1.16), we obtain a homomorphism from C_{SS}^{k} to H_{dR}^{k}(𝔛/K). The Main Theorem of [I-S] identifies it as the inverse of c^{SS}, thereby reducing the proof that Φ is an isomorphism, to the (known) fact that c^{SS} is an isomorphism.

1.5. Fourth description of the cohomology (through harmonic measures on the boundary). In a slightly different notation, this description appears in [S-S] already. Let B_{·} denote the profinite simplicial set of flags in V_{K}. Its k − 1 simplices (1 ≤ k ≤ d) consist of flags of length k

(1.21)  B_{k−1} = \{ \begin{matrix}W_{1} ⊃ ⋯ ⊃ W_{k}; W_{i} are proper subspaces of V_{K}, \\ properly contained in each other \end{matrix} \} ,

and the boundary map ∂ takes a flag to the alternating sum of the flags which are obtained from it by omittimg one of the subspaces. We view B_{k−1} as the space of K-points of a certain flag variety, which is compact and totally disconnected in the p-adic topology. It breaks into \binom{d}{k} components, labelled by the types of the flags. The component of minimal type B_{k−1}^{min} consists of flags where dim W_{i} = k + 1 − i.

Consider the space of K-valued distributions D(B_{k−1}^{min}, K). A distribution ν on B_{k−1}^{min} is called harmonic if either k = 1 and it is of total mass 0, or k ≥ 2 and ∂_{∗}ν = 0 (note that ∂ extends to a map ∂_{∗} : D(B_{k−1}, K) → D(B_{k−2}, K) and D(B_{k−1}^{min}, K) ⊂ D(B_{k−1}, K)). The harmonic distributions on B_{k−1}^{min} will be denoted D(B_{k−1}^{min}, K)_{har}. The space of harmonic measures on B_{k−1}^{min} is a K-Banach space in a natural way.

The lemma of Deligne-Schneider-Stuhler (Lemma 2.1 below) implies the isomorphism

(1.22)  D(A^{k+1}, K)/D(A^{k+1}, K)_{deg} ≃ D(B_{k−1}^{min}, K)_{har}.

We therefore obtain a fourth description of the cohomology, from the previous one.

1.6. Comparison of the various approaches. We are now ready to state our results. Call a flag of vector spaces

(1.23)  F = (W_{1} ⊃ ⋯ ⊃ W_{k})

compatible with an oriented k-simplex σ = (L_{0} ⊃ ⋯ ⊃ L_{k} ⊃ πL_{0}) if

(1.24)  L_{i} = πL_{0} + (W_{i} ∩ L_{0}).

Denote by B(σ) the open and compact subset in B_{k−1} of all the flags which are compatible with σ. Notice that if F is compatible with σ, their types coincide, and in particular F is of minimal type if and only if σ is. The B(σ) form a basis for the topology of B_{k−1} (compare [SS], Section 4, Proposition 8).

Theorem 1.1. Let [ω] ∈ H_{dR}^{k}(𝔛/K), and let c_{ω} ∈ C_{har}^{k} be the associated harmonic cochain. Then (see diagram in Section 2.2):

(i) There exists a unique harmonic distribution ν_{ω} ∈ D(B_{k−1}^{min}, K)_{har} satisfying

(1.25)  ν_{ω}(B(σ)) = c_{ω}(σ)

for every σ ∈ T̂_{k}^{min} (oriented k-simplex of minimal type).

(ii) If μ_{ω} ∈ D(A^{k+1}, K) corresponds to ν_{ω} under the Deligne-Schneider-Stuhler Lemma (1.22), then

(1.26)  Φ(μ_{ω}) = [ω].

(iii) Let λ_{σ} ∈ L(A^{k+1}) be the function λ_{σ}(S) = (σ, S). Then

μ_{ω}(λ_{σ}) = c_{ω}(σ).

(iv) Introduce coordinates as in Section 1.2. Then

(1.27)  c_{ω}(σ_{0}) = c_{ω}^{SS}(χ_{B_{I}})

where χ_{B_{I}} ∈ L(G/B_{I}) is the characteristic function of the principal coset B_{I}.

Part (iv) of the theorem implies that for any g ∈ G,

(1.28)  c_{ω}(gσ_{0}) = c_{ω}^{SS}(χ_{gB_{I}}).

Since G acts transitively on oriented k-simplices of a given type, c_{ω}^{SS} is the restriction of the harmonic cochain c_{ω} to cells of minimal type.

Contrary to the impression that (i) might have on the reader, it is not true that there is a distribution on all of B_{k−1} satisfying (1.25) for every σ. Equivalently, if B(σ) is the disjoint union of finitely many B(σ_{i}), unless the type of σ (and of the σ_{i}) is minimal, it is not true that (σ, S) = ∑(σ_{i}, S) for all S ∈ A^{k+1}. This “breaking of symmetry” among the types is built into the definition of C_{har}^{k}; see [dS], Section 3. It also manifests itself in the proof of Lemma 2.1 below, where the arguments work only for flags of minimal type.

Theorem 1.2. The following are equivalent.

(i) c_{ω} ∈ C_{bnd}^{k}

(ii) ν_{ω} is a measure

(iii) μ_{ω} can be taken to be a measure.

If this happens we call [ω] a bounded cohomology class. The norm of c_{ω}, the norm of ν_{ω}, and the Banach quotient norm of the class of μ_{ω}, all coincide. Furthermore, if we denote by Ω_{bnd}^{k} the image of (1.15), i.e. the space of all k-forms ω(μ) for μ ∈ M(A^{k+1}, K)/M(A^{k+1}, K)_{deg}, then the map ω ↦ [ω] is a G-equivariant isomorphism of Ω_{bnd}^{k} onto the space of bounded cohomology classes.

Conjecture 1.3. If Ω_{bnd}^{k} ⊂ Ω̃ ⊂ Ω_{cl}^{k} is any G-invariant subspace of closed k-forms which maps injectively into the cohomology (i.e. which intersects the exact forms trivially), then Ω_{bnd}^{k} = Ω̃.

Perhaps in the last conjecture one has to impose some topological restrictions on Ω̃. The passage from a bounded ν_{ω} to μ_{ω}, followed by (1.15), can be thought of as the p-adic Poisson integral. For k = d it coincides with the formulae of [S-T].

1.7. Example: d = k = 1. Here T is the famous tree, and B_{k−1}^{min} is ℙ(V_{K}), the set of ends of the tree. For an oriented edge σ, the set B(σ) consists of the ends of all the rays passing through σ. The function (σ, S) on S = (a_{0}, a_{1}) ∈ A^{2} gives S the value 1 if the geodesic from a_{0} to a_{1} passes through σ in the positive direction, −1 if it passes through σ in the negative direction, and 0 if it does not pass through σ. The relation between c_{ω} and ν_{ω} is well-known in this case. The formula for the canonical ω representing [ω] when ν_{ω} is bounded, is due to Teitelbaum [T]. Let us quickly recover it.

Start with a measure μ on A^{2}. Introduce the basis e_{0} = (1, 0) and e_{1} = (0, 1) of V_{K} = K^{2}. The corresponding measure ν on ℙ^{1}(K) = B_{k−1}^{min} is given by

ν(U) = μ {(e_{0}g, e_{1}g)| g^{−1}(0) ∈ U} − μ {(e_{0}g, e_{1}g)| g^{−1}(∞) ∈ U}

where g denotes an element of P GL_{2}(K). Note that if μ is degenerate, ν = 0, and that ν is harmonic, which in this case (k = 1) means that it is of total mass 0. An easy computation yields

ω(μ) = ∫_{A^{2}} ω(S)dμ(S) = ∫_{ℙ^{1}(K)} \frac{dz}{z − ζ} dν(ζ).

2. The proofs

2.1. The Lemma of Deligne-Schneider-Stuhler. Let F_{·}^{(k)} denote the profinite simplicial set of flags of subspaces of V_{K} whose dimensions do not exceed k ([S-S], p.66):

(2.1)  F_{r−1}^{(k)} = {V_{K} ⊇ W_{1} ⊇ ⋯ ⊇ W_{r}; 1 ≤ dim W_{i} ≤ k}

and let N F_{·}^{(k)} be the subset of non-degenerate flags (i.e. flags where W_{i} properly contains W_{i+1}). Note that N F_{r−1}^{(k)} is a subset of F_{r−1}^{(k)} (if we relax the assumption of non-degeneracy), but also of B_{r−1} (if we relax the restriction on the dimensions of the W_{i}), and as such

(2.2)  N F_{k−1}^{(k)} = B_{k−1}^{min}.

As explained in [S-S], p.66, the complex

(2.3)  0 → ℤ → L(F_{·}^{(k)})

computes the reduced cohomology of |F_{·}^{(k)}| with coefficients in ℤ, and since it is homotopically equivalent to the complex

(2.4)  0 → ℤ → L(N F_{·}^{(k)})

(loc.cit. p.69), we have

(2.5)  H̃^{k−1}(|F_{·}^{(k)}|, ℤ) = L(B_{k−1}^{min})/∂^{∗}L(N F_{k−2}^{(k)}),

with the understanding that if k = 1 the denominator is replaced by ℤ. Similarly the complex

(2.6)  0 → ℤ → L(Y_{·}^{(k)})

computes the cohomology of |Y_{·}^{(k)}|. If we denote by Y_{r} = A^{r+1} (without any condition on the dimension of the Span), Y_{·} is acyclic, and of course Y_{r} = Y_{r}^{(k)} for r ≤ k − 1. It follows that

H̃^{k−1}(|Y_{·}^{(k)}|, ℤ) = Ker(L(Y_{k−1}^{(k)}) → L(Y_{k}^{(k)}))/Im(L(Y_{k−2}^{(k)}) → L(Y_{k−1}^{(k)}))
= Ker(L(Y_{k−1}^{(k)}) → L(Y_{k}^{(k)}))/Ker(L(Y_{k−1}^{(k)}) → L(Y_{k}))
(2.7)  ≃ {f ∈ L(Y_{k}); f|_{Y_{k}^{(k)}} = 0 and ∂^{∗}f = 0}

where the last isomorphism is induced by ∂^{∗}, but this is just the subgroup that we denoted Λ′_{k}.

Lemma 2.1. For every k ≥ 1 there exists a G-equivariant isomorphism

(2.8)  L(B_{k−1}^{min})/∂^{∗}L(N F_{k−2}^{(k)}) ≃ Λ′_{k}

(the denominator on the left is equal to ℤ if k = 1), carrying χ_{B(σ_{0})}, the characteristic function of the open and compact set B(σ_{0}), to the function

(2.9)  λ_{σ_{0}} : S ↦ (σ_{0}, S)

(see Section 1.3).

Proof. Given (2.5) and (2.7), the existence of an isomorphism as above is a consequence of Proposition 5 of Section 3 of [S-S]. Let Z_{·,·}^{(k)} be the bisimplicial profinite set introduced in [S-S]. One has a commutative diagram (2.10)

·  |  ·
↑  |  ↑
L(F_{k−1}^{(k)}) | → L(Z_{k−1,0}^{(k)}) → ·
↑  |  ↑  ↑
⋯  |  ⋯  L(Z_{k−2,1}^{(k)}) → ·
↑  |  ↑
L(F_{1}^{(k)}) | → L(Z_{1,0}^{(k)}) → ·  ⋯  ·
↑  |  ↑ d′  ↑  ↑
L(F_{0}^{(k)}) | → L(Z_{0,0}^{(k)}) →^{d′′} L(Z_{0,1}^{(k)}) → ⋯ → L(Z_{0,k−1}^{(k)}) → ·
↑ | ↑ ↑ ↑
ℤ  → L(Y_{0}^{(k)}) → L(Y_{1}^{(k)}) → ⋯ → L(Y_{k−1}^{(k)}) → ·

Here the left column and the bottom row (which are separated from the double complex of the Z_{·,·}^{(k)} by dashes) are the complexes used to compute the reduced cohomologies H̃^{s}(|F_{·}^{(k)}|, ℤ) and H̃^{s}(|Y_{·}^{(k)}|, ℤ). These reduced cohomologies vanish for s < k−1 and we are interested in their value for s = k−1, which is why we wrote the k-th diagonal in the double complex (corresponding to Z_{r,s}^{(k)} with r + s = k − 1) in full.

The augmented rows and columns of the double complex are acyclic. This is the contents of Lemma 3, p.67 of [S-S], and is straightforward.

To compute the effect of the isomorphism

(2.11)  H̃^{k−1}(|F_{·}^{(k)}|, ℤ) ≃ H̃^{k−1}(|Y_{·}^{(k)}|, ℤ)

on the class of χ_{B(σ_{0})} ∈ L(F_{k−1}^{(k)}) we have to “go down the stairs” along the kth

diagonal in the diagram until we find the corresponding function χ_{Y} in L(Y_{k−1}^{(k)}) whose class [χ_{Y}] ∈ H̃^{k−1}(|Y_{·}^{(k)}|, ℤ) is the desired class. We then have to show (recall 2.7) that ∂^{∗}χ_{Y} ∈ L(Y_{k}) (which of course vanishes on Y_{k}^{(k)} and is therefore in Λ′_{k}), is the function S ⟼ (σ_{0}, S).

Let χ_{0}, …, χ_{k−1}, χ_{i} ∈ L(Z_{k−1−i,i}^{(k)}), be the functions constructed in the transition from χ_{B(σ_{0})} to χ_{Y}. Of course, there is some ambiguity in the process, as the

resulting element of L(Y_{k−1}^{(k)}) is only well-defined up to coboundaries, but we shall select nice candidates for the χ_{i}.

The first step gives that χ_{0} is the characteristic function of the set

(2.12)  Q_{k−1,0} = \{ \begin{matrix}(W_{1} ⊃ ⋯ ⊃ W_{k}; Ka_{k}) ∈ Z_{k−1,0}^{(k)}| \\ (W_{1} ⊃ ⋯ ⊃ W_{k} = Ka_{k}) is compatible with σ_{0} \end{matrix} \} .

Here Ka_{k} represents a line in V_{K} (a hyperplane H_{a_{k}} in V_{K}^{∗}) so a_{k} can be normalized to lie in L_{0} − πL_{0}, and then the condition that it spans W_{k} means that it lies in L_{k} − πL_{0}, i.e. ι_{σ_{0}}(a_{k}) = k. A simple computation reveals that χ_{0} = d′ψ_{0} is the coboundary (in the vertical direction) of the characteristic function ψ_{0} of

(2.13)  P_{k−2,0} = \{ \begin{matrix}(W_{1} ⊃ ⋯ ⊃ W_{k−1}; Ka_{k}) ∈ Z_{k−2,0}^{(k)}| \\ (W_{1} ⊃ ⋯ ⊃ W_{k−1} ⊃ Ka_{k}) is compatible with σ_{0} \end{matrix} \} .

Taking now the differential of ψ_{0} in the horizontal direction we see that χ_{1} = d′′ψ_{0} is the anti-symmetrization of the characteristic function of (2.14)

Q_{k−2,1} = \{ \begin{matrix}(W_{1} ⊃ ⋯ ⊃ W_{k−1}; Ka_{k−1}, Ka_{k}) ∈ Z_{k−2,1}^{(k)}| \\ (W_{1} ⊃ ⋯ ⊃ W_{k−1} = Ka_{k−1} + Ka_{k} ⊃ Ka_{k}) is compatible with σ_{0} \end{matrix} \} .

Here we use the following terminology. The symmetric group S_{m} acts on Z_{k−m,m−1}^{(k)}

by permuting the m hyperplanes. It therefore acts on L(Z_{k−m,m−1}^{(k)}) and the anti-

symmetrization of f ∈ L(Z_{k−m,m−1}^{(k)}) is the function f^{alt} defined by

(2.15)  f^{alt} = ∑_{π∈S_{m}} sgn(π) · ^{π}f.

Repeating the same argument, χ_{1} is the coboundary (in the vertical direction) of the anti-symmetrization of the characteristic function of (2.16)

P_{k−3,1} = \{ \begin{matrix}(W_{1} ⊃ ⋯ ⊃ W_{k−2}; Ka_{k−1}, Ka_{k}) ∈ Z_{k−3,1}^{(k)}| \\ (W_{1} ⊃ ⋯ ⊃ W_{k−2} ⊃ Ka_{k−1} + Ka_{k} ⊃ Ka_{k}) is compatible with σ_{0} \end{matrix} \}

and the (horizontal) differential of this function is χ_{2}, the anti-symmetrization of the characteristic function of (2.17)

Q_{k−3,2} = \{ \begin{matrix}(W_{1} ⊃ ⋯ ⊃ W_{k−2}; Ka_{k−2}, Ka_{k−1}, Ka_{k}) ∈ Z_{k−3,2}^{(k)}| \\ (W_{1} ⊃ ⋯ ⊃ W_{k−2} = Ka_{k−2} + Ka_{k−1} + Ka_{k} ⊃ ⋯ ⊃ Ka_{k}) \\ is compatible with & σ & 0 \end{matrix} \} .

At the end we arrive at χ_{k−1}, the anti-symmetrization of the characteristic function of the set (2.18)

Q_{0,k−1} = \{ \begin{matrix}(W_{1}; Ka_{1}, …, Ka_{k}) ∈ Z_{0,k−1}^{(k)}| \\ (W_{1} = Ka_{1} + ⋯ + Ka_{k} ⊃ ⋯ ⊃ Ka_{k}) is compatible with σ_{0} \end{matrix} \} .

We therefore see that χ_{Y} is the anti-symmetrization of the characteristic function of all those

(2.19)  (Ka_{1}, …, Ka_{k})

for which ι_{σ_{0}}(a_{j}) = j for each 1 ≤ j ≤ k.

Finally, it is easy to see that for S = (a_{0}, a_{1}, …, a_{k}),

(2.20)  ∂^{∗}χ_{Y} (S) = ∑_{i=0}^{k}(−1)^{i}χ_{Y} (a_{0}, …, â_{i}, …, a_{k}) = (σ_{0}, S).

This concludes the proof of the lemma.

Corollary 2.2. Let Λ_{k} ⊂ L(Y_{k}) be the ℤ-linear span of the functions λ_{σ} = (σ, −). Then Λ_{k} = Λ′_{k}.

Proof. One clearly has Λ_{k} ⊂ Λ′_{k} ⊂ L(Y_{k}). Note also that L(Y_{k})/Λ′_{k} is torsion-free, so for any field F

(2.21)  Λ′_{k} ⊗ F ⊂ L(Y_{k}) ⊗ F,

and Hom(Λ′_{k}, F) = D(Y_{k}, F)/D(Y_{k}, F)_{deg}. To prove the corollary it is enough to prove that for any field F, Λ_{k} ⊗ F = Λ′_{k} ⊗ F (indeed it is enough to check this for F = ℚ or 𝔽_{l}). If this were not the case, there would have been a non-zero μ ∈ Hom(Λ′_{k}, F) = D(Y_{k}, F)/D(Y_{k}, F)_{deg} vanishing on Λ_{k}, and in particular on the functions λ_{σ}. Let ν ∈ D(B_{k−1}^{min}, F)_{har} be the distribution on B_{k−1}^{min} which corresponds to μ under the lemma. Then

(2.22)  ν(B(σ)) = μ(λ_{σ}) = 0

for every σ of minimal type. As the B(σ) form a basis for the topology of B_{k−1}^{min}, ν = 0, hence μ = 0 as well. This contradiction shows that the ℤ-span of the λ_{σ} is precisely the kernel of (ι^{∗}, ∂^{∗}), as desired.

2.2. Proof of Theorem 1.1. In [dS] we showed that C_{har}^{k} is the algebraic dual of Λ_{k} ⊗ K, and that

(2.23)  ⟨λ_{σ}, c_{ω(S)}⟩ = c_{ω(S)}(σ) = (σ, S).

From the lemma and its corollary we get, upon dualizing, the equalities

(2.24)  D(B_{k−1}^{min}, K)_{har} ≃ D(A^{k+1}, K)/D(A^{k+1}, K)_{deg} ≃ C_{har}^{k}

as in (i) (ii) and (iii) of Theorem 1.1. See the commutative diagram below, in which Ψ is dual to the identity Λ_{k} = Λ′_{k}. Since two sides of the diagram are isomorphisms, so is the third, Φ.

H_{dR}^{k}(𝔛/K)
[2.1]  [I-S] Φ ↗
D(B_{k−1}^{min}, K)_{har} ≃ D(A^{k+1}, K)/D(A^{k+1}, K)_{deg}  ↓ ω ↦ c_{ω}
Ψ ↘
C_{har}^{k}

On the other hand the Main Theorem of [I-S], together with Lemma 2.1, imply the formula

(2.25)  c_{ω(S)}^{SS}(χ_{B_{I}}) = (σ_{0}, S),

which gives part (iv) of the theorem.

2.3. Proof of Theorem 1.2. If instead of taking Hom(−, K) in the proof of theorem 1.1, we take Hom(−, O_{K}), one gets the first part of theorem 1.2 (the equality of the norms on the bounded cohomology in the three interpretations). As for the second simply note, as in [I-S], that the isomorphism

(2.26)  Φ : M(A^{k+1}, K)/M(A^{k+1}, K)_{deg} ≃ H_{dR}^{k}(𝔛/K)_{bnd}

factors through Ω_{bnd}^{k} (see 1.15).

3. The cohomology of Γ\𝔛

3.1. A Hodge-like decomposition of the cohomology. Let Γ be a discrete cocompact subgroup of G. For simplicity make the assumption that if 1 ≠ γ ∈ Γ, then for every vertex v of T , the distance between v and γv is at least 3. This assumption is needed in order to make Γ\T into a simplicial complex, and can be met by passing to a subgroup of finite index^{1}. By a theorem of Mustafin [Mu], Γ\𝔛 is the rigid analytic space associated to a unique smooth projective variety X_{Γ}, defined over K. Let π_{Γ} : 𝔛 → X_{Γ} be the canonical projection. For τ ∈ T_{r} let St(τ) (the star of τ) be the union of all the (open) simplices of |T | containing τ. Define

(3.1)  X(τ) := r^{−1}(St(τ)) = ⋂_{v∈τ} X(v)

and X_{Γ}(τ) := π_{Γ}(X(τ)). Then U = {X(v); v ∈ T_{0}} and U_{Γ} = {X_{Γ}(v); v ∈ Γ\T_{0}} are admissible coverings of 𝔛 and X_{Γ} respectively, whose nerves are just T and Γ\T .

There are two spectral sequences abutting to the de-Rham cohomology H_{dR}^{n}(X_{Γ}). (Note that by GAGA, analytic and algebraic de-Rham cohomology coincide.) The first is the covering spectral sequence

(3.2)  ′E_{2}^{r,s} = H^{r}(Γ, H_{dR}^{s}(𝔛)) ⇒ H_{dR}^{n}(X_{Γ}).

The associated filtration is denoted by F_{Γ}^{·}H_{dR}^{n}(X_{Γ}). The second is the Hodge-to-de-Rham spectral sequence

(3.3)  ′′E_{1}^{r,s} = H^{s}(X_{Γ}, Ω^{r}) ⇒ H_{dR}^{n}(X_{Γ}),

and the associated filtration is the Hodge filtration F_{dR}^{·}H_{dR}^{n}(X_{Γ}). Below, we shall construct them as the two spectral sequences attached to a certain double complex (see also [dS], Section 9).

Our purpose is to show how the isomorphism Ω_{bnd}^{s} ≃ H_{dR}^{s}(𝔛)_{bnd} ≃ C_{bnd}^{s} and the results of [dS] yield the following theorem.

1Our condition guarantees that if (v_{0}, …, v_{r}) ∈ T_{r}, and γ_{i} ∈ Γ, then the γ_{i}v_{i} are distinct, and if (γ_{0}v_{0}, …, γ_{r}v_{r}) ∈ T_{r}, then all the γ_{i} are equal. Defining (Γ\T )_{r} = Γ\T_{r} makes Γ\T a simplicial complex, and |Γ\T | = Γ\|T |.

Theorem 3.1. (i) The covering spectral sequence degenerates at ′E_{2}.

(ii) The Hodge-to-de-Rham spectral sequence degenerates at ′′E_{1}.

(iii) The two filtrations are opposite to each other: For every 0 ≤ n ≤ d, and every p

(3.4)  H_{dR}^{n}(X_{Γ}) = F_{Γ}^{n+1−p}H^{n} ⊕ F_{dR}^{p}H^{n}.

As a corollary, if we put, for p + q = n,

(3.5)  H^{q,p} = F_{Γ}^{q}H^{n} ∩ F_{dR}^{p}H^{n},

we have a Hodge-like direct sum decomposition

(3.6)  H^{n} = ⊕H^{q,p}.

The three statements of the theorem are not new. Part (i) was proved by Schneider in [S]. Part (ii) is classical, and is usually proved by complex Hodge theory (the theory of harmonic forms on Kähler manifolds). Our substitute for the complex theory is harmonic analysis on the building, and as far as we know, this constitutes a new proof (for p-adically uniformized varieties). Part (iii) was conjectured by Schneider in [S], and proved by Iovita and Spiess in [I-S]. Our proof is closer in spirit to Section 9 of [dS], but the key idea, namely the existence of a Γ-module Ω_{bnd}^{s} mapping isomorphically onto the bounded cohomology, is the same.

3.2. Generalities on double complexes. We recall the two spectral sequences attached to a double complex. See [Go] I.4.8 for the details. Let K^{r,s} be a double complex of vector spaces vanishing for r < 0 or s < 0. We denote the differentials of the complex by

(3.7)  d′ : K^{r,s} → K^{r+1,s} d′′ : K^{r,s} → K^{r,s+1} d′d′′ + d′′d′ = 0.

The total complex is

(3.8)  tot(K)^{n} = ⊕_{r+s=n}K^{r,s}, D = d′ + d′′.

There are two spectral sequences abutting to the homology H^{n} = h^{n}(tot(K)^{·}, D). The first spectral sequence starts with

(3.9)  ′E_{1}^{r,s} = h^{s}(K^{r,·}, d′′)

and ′d_{1}^{r,s} : ′E_{1}^{r,s} → ′E_{1}^{r+1,s} is derived from d′. The induced filtration on the abuttment is denoted ′F^{·}H^{n}. The second spectral sequence is obtained by reversing the roles of the indices. Thus

(3.10)  ′′E_{1}^{r,s} = h^{s}(K^{·,r}, d′),

where ′′d_{1}^{r,s} is derived from d′′, and the induced filtration is denoted by ′′F^{·}H^{n}.

A map of complexes induces maps between the (first or second) spectral sequences, and maps on the abuttments, compatible with both filtrations.

3.3. A trivial but useful example. Suppose that d′′ = 0 throughout. Then putting H^{r,s} = h^{r}(K^{·,s}, d′)

(3.11)  H^{n} = ⊕_{r+s=n}H^{r,s}.

The first spectral sequence gives ′E_{1}^{r,s} = K^{r,s}, and degenerates at ′E_{2}^{r,s} = H^{r,s} (i.e. ′d_{2}^{r,s} = ′d_{3}^{r,s} = ⋯ = 0). For the corresponding filtration we find

(3.12)  ′F^{q}H^{n} = ⊕_{r≥q}H^{r,n−r}.

On the other hand the second spectral sequence degenerates at ′′E_{1}^{r,s} = H^{s,r} and

(3.13)  ′′F^{p}H^{n} = ⊕_{r≥p}H^{n−r,r}.

In particular the two filtrations are opposite to each other, and if p + q = n,

(3.14)  ′F^{q}H^{n} ∩ ′′F^{p}H^{n} = H^{q,p}.

Lemma 3.2. Let φ : K̃^{r,s} → K^{r,s} be a map of double complexes. Assume that for K̃^{r,s}, we have d̃′′ = 0 throughout.

(i) Assume that the maps ′E_{2}φ : ′Ẽ_{2}^{r,s} → ′E_{2}^{r,s} are isomorphisms. Let H̃^{n} and H^{n} be the homologies of the total complexes. Then the induced map Hφ : H̃^{n} → H^{n} is an isomorphism for every n, under which ′F̃^{·}H̃^{n} and ′F^{·}H^{n} agree (i.e. map isomorphically to each other), and the spectral sequence ′E_{2}^{r,s} ⇒ H^{n} degenerates at ′E_{2}.

(ii) If furthermore ∞ > dim ′′Ẽ_{1}^{r,s} ≥ dim ′′E_{1}^{r,s} for every r and s, then ′′E_{1}φ : ′′Ẽ_{1}^{r,s} → ′′E_{1}^{r,s} is an isomorphism, the filtrations ′′F̃^{·}H̃^{n} and ′′F^{·}H^{n} agree, and the spectral sequence ′′E_{1}^{r,s} ⇒ H^{n} degenerates at ′′E_{1}.

Proof. (i) Since the first spectral sequence of the double complex K̃^{·,·} degenerates at ′Ẽ_{2}, we get under the assumption that ′E_{2}φ is an isomorphism, that the same is true for the complex K^{·,·}. Furthermore, the map Hφ on the abuttments induces isomorphisms on the graded pieces with respect to the first filtration, and is therefore an isomorphism of filtered vector spaces.

(ii) Since ′′Ẽ_{1}^{r,s} = ′′Ẽ_{∞}^{r,s}, a dimension count shows that we must have ′′E_{1}^{r,s} = ′′E_{∞}^{r,s} as well, so the second spectral sequence of the complex K^{·,·} degenerates at′′E_{1} too. The same dimension count shows that we must have an equality dim ′′Ẽ_{1}^{r,s} = dim ′′E_{1}^{r,s}. The map Hφ is an isomorphism of vector spaces, compatible with the second filtration on H̃^{n} and H^{n}. As we have just remarked, the graded pieces with respect to the second filtration have the same dimensions. Thus Hφ is an isomorphism of filtered vector spaces also with respect to the second filtration.

Corollary 3.3. Under the conditions (i) and (ii) of the lemma, the filtrations ′F^{·} and ′′F^{·} on H^{n} are opposite to each other.

3.4. The double complex K^{·,·}. We repeat some of the arguments of [dS], Section 9. Introduce the double complex

(3.15)  K^{r,s} = C^{r}(U_{Γ}, Ω^{s}) = C^{r}(U, Ω^{s})^{Γ}

of Čech cochains for the cover U_{Γ} with values in the presheaf of rigid analytic forms. Thus an element of C^{r}(U, Ω^{s}) is a map f which assigns to every τ ∈ T̂_{r} an element f(τ) ∈ Ω^{s}(X(τ)). The action of γ ∈ Γ is given as usual by the rule (γf)(τ) = γ(f(γ^{−1}τ)), and K^{r,s} is by definition the subspace of Γ-invariants. The differentials are d′ = δ (Čech differntiation) and d′′ = (−1)^{r}d. The total complex tot(K)^{·} computes the Čech hyper-cohomology Ȟ^{n}(U_{Γ}, Ω^{·}). Since the open sets of the cover and all their intersections are Ω^{·}-acyclic, we find that the homology of the total complex is

(3.16)  H^{n} = H^{n}(X_{Γ}, Ω^{·}) = H_{dR}^{n}(X_{Γ}).

Let us compute the two spectral sequences attached to this double complex. The first is

(3.17)  ′E_{1}^{r,s} = C^{r}(U_{Γ}, \underline{H}_{dR}^{s}).

Here \underline{H}_{dR}^{s} is the presheaf on U of (rigid) de-Rham cohomology. As explained in [dS], it is isomorphic to the presheaf denoted there by \underline{A}^{s}. Now the acyclicity theorem ([dS], Theorem 5.7) claims that the Čech complex

(3.18)  0 → C_{har}^{s} → C^{·}(U, \underline{A}^{s})

is an acyclic resolution of the Γ-module of harmonic s-cochains on the building. We therefore find that

(3.19)  ′E_{2}^{r,s} = Ȟ^{r}(U_{Γ}, \underline{H}_{dR}^{s}) = h^{r}(C^{·}(U, \underline{A}^{s})^{Γ}, δ) = H^{r}(Γ, C_{har}^{s}).

Thus the first spectral sequence is nothing but the covering spectral sequence

(3.20)  H^{r}(Γ, H_{dR}^{s}(𝔛)) ⇒ H_{dR}^{n}(X_{Γ}).

The second spectral sequence is, by definition, the Hodge-to-de-Rham spectral sequence

(3.21)  ′′E_{1}^{r,s} = Ȟ^{s}(U_{Γ}, Ω^{r}) = H^{s}(X_{Γ}, Ω^{r}) ⇒ H_{dR}^{n}(X_{Γ}).

3.5. The double complex K̃^{·,·}, and the end of the proof. Consider now the subcomplex K̃^{r,s} ⊂ K^{r,s}

(3.22)  K̃^{r,s} = C^{r}(U_{Γ}, Ω_{bnd}^{s}) = C^{r}(U, Ω_{bnd}^{s})^{Γ},

of Čech cocycles with values in the locally constant system of global bounded sforms. Thus an element of C^{r}(U, Ω_{bnd}^{s}) is a function f which assigns to each τ ∈ T̂_{r} an s-form f(τ) ∈ Ω_{bnd}^{s} ⊂ Ω^{s}(X(τ)), and f is Γ-invariant if f(γ(τ)) = γf(τ) for every γ ∈ Γ. Since bounded forms are closed, d′′ = 0 on K̃^{r,s}.

If we take for φ the inclusion of K̃^{r,s} in K^{r,s} we find ourselves in the situation of Lemma 3.2. In order to apply it, one has to check conditions (i) and (ii). Recalling the isomorphism Ω_{bnd}^{s} ≃ C_{bnd}^{s} we see that

(3.23)  ′Ẽ_{2}^{r,s} = Ȟ^{r}(U_{Γ}, C_{bnd}^{s}) ≃ H^{r}(Γ, C_{bnd}^{s}).

The last isomorphism stems from the fact that |T | is a contractible space on which Γ acts, and the nerve of U_{Γ} is Γ\T . Condition (i) boils down now to the following lemma.

Lemma 3.4. The inclusion of C_{bnd}^{s} in C_{har}^{s} induces an isomorphism in group cohomology

(3.24)  H^{r}(Γ, C_{bnd}^{s}) ≃ H^{r}(Γ, C_{har}^{s}).

Proof. The proof is based on the acyclicity theorem ([dS], Theorem 5.7). The Γmodule C_{bnd}^{s}(O_{K}) of the integral-valued harmonic s-cochains, admits an acyclcic resolution

(3.25)  0 → C_{bnd}^{s}(O_{K}) → C^{·}(U, \underline{A}^{s}(O_{K}))

similar to the one for C_{har}^{s}. To deduce it from [dS] observe that the arguments there yield the wanted version with integral coefficients because C_{bnd}^{s}(O_{K}) = Hom(Λ_{k}, O_{K}), and because theorem 5.3 of [dS] holds for Λ_{k} with coefficients in ℤ, as we take it to be in this work, even before we extend scalars to K.

We therefore have

(3.26)  H^{r}(Γ, C_{bnd}^{s}(O_{K})) = h^{r}(C^{·}(U, \underline{A}^{s}(O_{K}))^{Γ}).

Since Γ is finitely generated and U_{Γ} is finite,

H^{r}(Γ, C_{bnd}^{s}) = h^{r}(C^{·}(U, \underline{A}^{s}(O_{K}))^{Γ} ⊗_{O_{K}} K)
(3.27)  = h^{r}(C^{·}(U, \underline{A}^{s}(K))^{Γ}) = H^{r}(Γ, C_{har}^{s}).

The second point to check is condition (ii) of Lemma 3.2. But we have already seen that ′′E_{1}^{r,s} = H^{s}(X_{Γ}, Ω^{r}), and from the last lemma ′′Ẽ_{1}^{r,s} = H^{s}(Γ, C_{har}^{r}) ≃ H^{s}(Γ, H_{dR}^{r}(𝔛)). The dimensions of these two pieces have been computed by Schneider in [S]:

(3.28)  dim H^{s}(X_{Γ}, Ω^{r}) = dim H^{s}(Γ, H_{dR}^{r}(𝔛)) = μ(Γ)δ_{r+s,d} + δ_{r,s}

where μ(Γ) = dim H^{d}(Γ, K).

Lemma 3.2 and corollary 3.3 now conclude the proof of Theorem 3.1. Incidentally note that the isomorphism ′′Ẽ_{1}^{r,s} ≃ ′′E_{1}^{r,s} is the isomorphism

(3.29)  gr_{Γ}^{s}H^{n} = H^{s,r} = gr_{dR}^{r}H^{n}.

References

[dS] de Shalit, E. : Residues on buildings and de Rham cohomology of p-adic symmetric domains, Duke Math. J., 106, 123-191 (2000)

[Go] Godement, R.: Topologie algebrique et théorie des faisceaux. Hermann, Paris (1973)

[I-S] Iovita, A., Spiess, M.: Logarithmic differential forms on p-adic symmetric spaces, preprint, January 2000.

[Mus] Mustafin, G.A. : Nonarchimedean uniformization, Math USSR Sbornik 34, 187-214 (1978)

[S] Schneider, P. : The cohomology of local systems on p-adically uniformized varieties, Math. Ann. 293, 623-650 (1992)

[S-S] Schneider, P., Stuhler, U. : The cohomology of p-adic symmetric spaces, Inv. Math. 105, 47-122 (1991)

[S-T] Schneider, P., Teitelbaum, J. : An integral transform for p-adic symmetric spaces, Duke Math. J. 86, 391-433 (1997)

[T] Teitelbaum, J. : Values of p-adic L-functions and a p-adic Poisson kernel. Inv. Math. 101, 395-410 (1990)