On the pro-étale cohomology of quotient stacks of Drinfeld spaces
Zecheng Yi
TL;DR
We compute explicit $ ext{ell}$-adic and $p$-adic pro-étale cohomology for quotient stacks of Drinfeld spaces by $ ext{GL}_n(\\mathcal{O}_K)$ and by $ ext{GL}_n(K)$, revealing their moduli interpretations as stacks of special formal $\\mathcal{O}_D$-modules. The core method uses the Scholze–Weinstein isomorphism between the Drinfeld and Lubin–Tate towers and the Colmez–Dospinescu–Nizioł descriptions of pro-étale cohomology, combined with condensed mathematics to endow a natural topology on cohomology groups. The main results give precise $ ext{ell}$-adic and $p$-adic cohomology of the isomorphism and isogeny stacks, and an application computes continuous cohomology of $ ext{GL}_n(\\mathbb{Q}_p)$ on dual generalized Steinberg representations, leading to a shifting pattern $H^{*}_{cts}( ext{GL}_n(\\mathbb{Q}_p), \mathrm{Sp}_r(\mathbb{Q}_p)^*) \cong \Lambda_{\mathbb{Q}_p}(x,y,x_3,\ldots,x_{2n-1})[-r]$ for $0\le r\le n-1$. This work thus connects period-domain cohomology, moduli interpretations of isomorphism/isogeny stacks, and representation-theoretic cohomology through a cohesive use of perfectoid/condensed techniques.
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-étale 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-étale 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 $\mathbb{Q}_p$.