On the pro-étale cohomology of quotient stacks of Drinfeld spaces

Let ${\mathcal{H}}_K^{n-1}$ denote the $(n-1)$-dimensional Drinfeld space over a $p$-adic field $K$. We give an explicit description of the $\mathrm{\ell }$-adic and $p$-adic pro-étale cohomology of quotient stacks $[{\mathcal{H}}_K^{n-1}\mathrm{/}{\mathrm{GL}}_n({\mathcal{O}}_K)]$ and $[{\mathcal{H}}_K^{n-1}\mathrm{/}{\mathrm{GL}}_n(K)]$, which are moduli stacks of special formal ${\mathcal{O}}_D$-modules. The computation makes use of the isomorphism between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze–Weinstein, as well as the $p$-adic pro-étale cohomology of the Drinfeld spaces computed by Colmez–Dospinescu–Niziol. As an application, we also compute the continuous group cohomology of ${\mathrm{GL}}_n({\mathbb{Q}}_p)$ over duals of generalized Steinberg representations over ${\mathbb{Q}}_p$.