Gluing sheaves along Harder-Narasimhan strata of $\mathrm{Bun}_G$
Jon Miles
TL;DR
The paper develops a geometric approach to gluing prime-to-$p$ torsion sheaves along Harder–Narasimhan strata of Bun$_G$ by relating excision gluing to the cohomology of smooth covers $ ilde{ ext{M}}_b$, and it connects these constructions to geometric constant-term functors and Eisenstein series. For general $G$, it formalizes how $i_b^*$ is realized via the cohomology of $ ilde{ ext{M}}_b$ and describes how the gluing data recovers an explicit tower of $v$-stacks whose cohomology encodes constant-term-type information. The GL$_2$ case is treated in depth, with explicit computations for half-integral and integral slopes, yielding concrete descriptions of $i_{b_2}^*Ri_{b_1,*}$ on various smooth representations, including unramified characters, principal series, and cuspidals, and demonstrating purity and vanishing phenomena that mirror Jacquet-type behavior in the automorphic side. The results illuminate the automorphic content of the Fargues–Scholze program by providing explicit geometric realizations of constant terms, Jacquet modules, and Eisenstein-series-like gluing in a concrete, computable setting. Overall, the work advances a deeper, computable bridge between the geometry of $v$-stacks, Banach–Colmez spaces, and the smooth representation theory of $p$-adic groups through the Harder–Narasimhan stratification of Bun$_G$.
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 $P_b$.