Residues on buildings and de Rham cohomology of \(p\)-adic symmetric domains.
For a finite extension \(K\) of the field \(\mathbb{Q}_p\), Drinfeld's \(p\)-adic symmetric domain of dimension \(d\) over \(K\) is defined as the complement \({\mathcal X}\) in \(\mathbb{P}^d\) of the union of all the \(K\)-rational hyperplanes. Endowed with the natural structure of a rigid analytic space, this domain \({\mathcal X}\) is acted on by the group \(G:=\text{PGL}_{d+1}(K)\), and its central role in both the study of Shimura varieties and the representation theory of \(G\) has become manifest during the past three decades. However, in contrast to the classical real symmetric domains, \(p\)-adic symmetric domains are not simply connected in the étale topology, which makes their cohomology theories difficult to tackle. In a fundamental paper published in 1991, \textit{P. Schneider} and \textit{U. Stuhler} [Invent. Math. 105, 47--122 (1991; Zbl 0751.14016)] showed how to compute the cohomology of \({\mathcal X}\) for any cohomology theory satisfying certain additional axioms, including the rigid de Rham cohomology. Their approach established, for each \(0\leq k\leq d\), an isomorphism between \(H^k_{\text{dR}}({\mathcal X})\) and a certain space \(C^k\) of \(k\)-cochains on the Bruhat-Tits building \({\mathcal T}\) of \(G\). In the paper under review, the author re-examines this cohomological result by Schneider and Stuhler, together with a number of questions left open back then. In fact, the author developes a considerably modified, somewhat more explicit and combinatorial approach to computing the de Rham cohomology of \(p\)-adic symmetric domains. To this end, he first analyzes local systems on Bruhat-Tits buildings attached to hyperplane arrangements, describes the related harmonic analysis and the corresponding simplicial cohomology, and then uses this explicit description to develop a novel theory of residues on Bruhat-Tits buildings in arbitrary dimension. The notion of residue of a closed form along simplices in a Bruhat-Tits building, combined with a detailed combinatorial study of the ring of associated logarithmic classes, finally leads to a new interpretation of the cohomology of a \(p\)-adic symmetric domain \({\mathcal X}\), which not only recovers the main theorem of Schneider and Stuhler, but also answers (in the affirmative) some of the open questions originating from their work. More precisely, the author's main theorem states that his so-called residue cochain map establishes an isomorphism of \(G\)-modules \(H^k_{\text{dR}} ({\mathcal X})\cong C^k_{\text{har}}\), where \(C^k_{\text{har}}\) denotes the space of all harmonic \(k\)-cochains on the related Bruhat-Tits building \({\mathcal T}\). In this context, the space \(C^k\) in the isomorphism theorem of Schneider and Stuhler is then recovered by restricting the author's harmonic \(k\)-cochains to cells of minimal type. The present very detailed paper is enhanced by a fairly extensive appendix on basic facts (with full proofs) concerning rigid de Rham cohomology theory, which is very useful purely as such.
scientific article; zbMATH DE number 1820775
scientific article