Splitting and making explicit the de Rham complex of the Drinfeld space

Abstract

Let $p$ be a prime number, $K$ a finite extension of ${\mathbb{Q}}_p$ and $n$ an integer $\ge 2$. We completely and explicitly describe the global sections ${\Omega }^{\bullet }$ 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 ${\Omega }^{\bullet }\twoheadrightarrow H^{n-1}({\Omega }^{\bullet })[-(n-1)]$.

289 pages, 13 figures. Minor modifications after the referee report, including an correction in the title