Cohomology of discrete groups in harmonic cochains on buildings
type: Item
instance of: scholarly article
title: Cohomology of discrete groups in harmonic cochains on buildings
zbMATH Open document ID: 1073.14027
DOI: 10.1007/BF02776064
author: Gil Alon
author: Ehud de Shalit
published in: Israel Journal of Mathematics
publication date: 2004-03-08T00:00:00Z
review text: 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{\mathca…
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 20E42
zbMATH DE Number: 2053071
zbMATH Keywords: Bruhat-Tits building
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02776064
OpenAlex ID: W2062843654
description: scientific article; zbMATH DE number 2053071