Splitting and making explicit the de Rham complex of the Drinfeld space
Christophe Breuil, Zicheng Qian
TL;DR
The paper provides an explicit, representation-theoretic splitting of the de Rham complex $\Omega^\bullet$ of the Drinfeld space in dimension $n-1$ into $GL_n(K)$-representations, realized as duals of locally $K$-analytic $G$-modules and organized as finite-length coadmissible $D(G)$-modules. It constructs explicit building blocks $X_k$ and $\widetilde{\Omega}^k$ to describe the internal structure of $\Omega^k$, and proves the existence of a canonical section in the derived category to the top-degree cohomology $H^{n-1}(\Omega^\bullet)$. The methodology hinges on a Lie-algebra/smooth-ext decomposition via a ST05-type spectral sequence, translation and wall-crossing functors, and detailed Weyl-group combinatorics (including Bernstein–Zelevinsky theory and Orlik–Strauch technology). The results yield a full, explicit splitting of the de Rham complex for general $n$, with precise control of extensions and filtrations, and they connect to $p$-adic Langlands through connections with Orlik–Strauch representations and potential links to completed cohomology. The work thus advances explicit realizations of $p$-adic geometric representations and provides tools for future Langlands-type correspondences in the locally analytic setting.
Abstract
Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $n$ an integer $\geq 2$. We completely and explicitly describe the global sections $Ω^\bullet$ of the de Rham complex of the Drinfeld space over $K$ in dimension $n-1$ as a complex of (duals of) locally $K$-analytic representations of $\mathrm{GL}_n(K)$. Using this description, we construct an explicit section in the derived category of (duals of) finite length admissible locally $K$-analytic representations of $\mathrm{GL}_n(K)$ to the canonical morphism of complexes $Ω^\bullet \twoheadrightarrow H^{n-1}(Ω^\bullet)[-(n-1)]$.