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

Abstract: Let $\mathcal{H}^{n-1}_{K}$ denote the $(n-1)$-dimensional Drinfeld space over a $p$-adic field $K$. We give an explicit description of the $\ell$-adic and $p$-adic pro-\'etale cohomology of quotient stacks $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(\mathcal{O}_K)]$ and $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(K)]$, 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-\'etale cohomology of the Drinfeld spaces computed by Colmez--Dospinescu--Niziol. As an application, we also compute the continuous group cohomology of $\operatorname{GL}_n(\mathbb{Q}_p)$ over duals of generalized Steinberg representations over $\mathbb{Q}_p$.