Cohomology of discrete groups in harmonic cochains on buildings (Q1424047)

Let \(K\) be a finite extension of \({{\mathbb Q}_p}\), and \(\varpi\) a uniformiser. Let \(V_K\) be a \((d+1)\)-dimensional vector space, and let \(G\) be \({\text{PGL}}(V_K)\). Let \({\mathcal T}\) denote the Bruhat-Tits building, that is, the simplicial complex in which the vertices are the sets \(\{[L_0],[L_1],\ldots,[L_k]\}\) where \(L_0\supset L_1\supset\cdots\supset L_k\supset\varpi L_0\) (strict inclusions). These inclusions determine a canonical cyclic ordering of the vertices in any simplex. Let \(\hat{\mathcal T}\) be the oriented version of the building, in which the \(k\)-simplices are the sequences \(\sigma=([L_0],[L_1],\ldots,[L_k])\). The cyclic group \(Z_{k+1}\) acts on the set \(\hat{\mathcal T}_k\) of oriented \(k\)-simplices by cyclically permuting the vertices. Set \({\mathcal A}={\mathbb P}(V_K)\) (viewed as a compact, totally disconnected space under the \(p\)-adic topology, equipped with a continuous \(G\)-action). One defines a pairing \[ (\;,\;)\colon\hat{\mathcal T}_k\times{\mathcal A}^{k+1}\to\{-1,0,1\} \] in the following way: For \(\sigma=([L_0],[L_1],\dots,[L_k])\in\hat{\mathcal T}_k\) and \(S=([a_0],[a_1],\dots,[a_k])\in {\mathcal A}^{k+1}\), where the representatives \(a_i\) are chosen so that \(a_i\in L_0\backslash\varpi L_0\), \[ (\sigma,S):={\text{sgn}}(\pi), \] if there exists a permutation \(\pi\) of \(\{0,1,\dots,k\}\) such that \(a_{\pi(i)}\in L_i\backslash L_{i+1}\) (where \(L_{k+1}:=\varpi L_0\)), and \[ (\sigma,S):=0 \quad\text{otherwise.} \] Let \(L({\mathcal A}^{k+1})=C^\infty({\mathcal A}^{k+1},\mathbb Z)\) be the abelian group of locally constant functions from \({\mathcal A}^{k+1}\) to \(\mathbb Z\). For \(\sigma\in\hat{\mathcal T}_k\), we denote by \(\lambda_\sigma\) the element of \(L({\mathcal A}^{k+1})\) defined by \[ \lambda_\sigma(S):=(\sigma,S), \] for any \(S\in{\mathcal A}^{k+1}\). Let \(\Lambda_k\subset L({\mathcal A}^{k+1})\) denote the span of all the functions \(\lambda_\sigma\) for \(\sigma\in\hat{\mathcal T}_k\). For a field \(F\), the space of \(F\)-valued harmonic \(k\)-cochains \(C_{\text{har}}^k=C_{\text{har}}^k(F)\) is then defined as \[ C_{\text{har}}^k:=\Hom(\Lambda_k,F). \] The first goal of the paper is to construct canonical extensions of \(G\)-modules \[ 0\longrightarrow C_{\text{har}}^{k-1}\longrightarrow {\tilde C}_{\text{har}}^{k-1}\longrightarrow C_{\text{har}}^k,\leqno(*) \] for every \(1\leq k\leq d\) (this is done by constructing extensions of \(G\)-modules \[ 0\longrightarrow \Lambda_k\longrightarrow \widetilde\Lambda_{k-1}\longrightarrow \Lambda_{k-1}, \] and dualizing them), and to describe \({\tilde C}_{\text{har}}^{k-1}\) by means of certain harmonicity conditions on the \((k-1)\)-cochains on the building. The second goal of the paper is to examine the cohomology of a discrete cocompact subgroup \(\Gamma\) of \(G\), with values in these spaces. One may assume without loss of generality that \(\Gamma\) is torsion free, and acts freely on \(\hat{\mathcal T}\). One assume that the characteristic of \(F\) is \(0\). Then, for \(1\leq k\leq d\) and \(r\geq 0\), let \[ \nu\colon H^r(\Gamma,C_{\text{har}}^k)\to H^{r+1}(\Gamma,C_{\text{har}}^{k-1}) \] be the connecting homomorphism coming from the extension~\((*)\). The authors prove that, if \(r+k=d\), then \(\nu\) is an isomorphism, except when \(r=k-2\), in which case \(\nu\) is injective, or \(r=k\), in which case it is surjective.