Gluing sheaves along Harder-Narasimhan strata of $\mathrm{Bun}_G$

TL;DR

This paper develops a method to glue torsion sheaves along Harder-Narasimhan strata of the moduli stack of $G$-bundles, using cohomology of strata in smooth charts, with explicit results for $GL_2$.

Contribution

It introduces a new approach to gluing sheaves along stratifications of $ ext{Bun}_G$ via cohomological techniques, extending the understanding of geometric constant term functors.

Findings

Explicit descriptions of images of geometric constant term functors for $GL_2$.

A new cohomological framework for sheaf gluing along Harder-Narasimhan strata.

Connections between stratification cohomology and moduli of split parabolic bundles.

Abstract

We describe how to glue prime-to-$p$ torsion sheaves along Harder-Narasimhan strata of Fargues-Scholze's $\mathrm{Bun}_G$ in terms of the cohomology of locally closed strata inside the smooth charts $\mathcal{M}_b \to \mathrm{Bun}_G$ constructed in [FS21], which are moduli of certain split parabolic bundles. Our computations for $G=\operatorname{GL}_2$ explicitly describe images of geometric constant term functors when restricted to a distinguished point inside $\mathrm{Bun}_{T_{\{b\}}}$, where $T_{\{b\}}$ is the split inner form of the Levi quotient of the corresponding parabolic subgroup $P_b$.