Automorphic Gluing

TL;DR

This paper establishes a gluing theorem on the automorphic side of the geometric Langlands correspondence, linking tempered and non-tempered categories across different groups, and proves related functorial properties.

Contribution

It proves a new automorphic gluing theorem that aligns with the spectral side, and demonstrates functorial preservation of tempered objects by key functors.

Findings

The difference between DMod(Bun_G) and its tempered subcategory is balanced by categories for Levi subgroups.

Certain functors preserve tempered objects, while others preserve anti-tempered objects.

Abstract

We prove a gluing theorem on the automorphic side of the geometric Langlands correspondence: roughly speaking, we show that the difference between $\mathrm{DMod}(\mathrm{Bun}_G)$ and its full subcategory $\mathrm{DMod}(\mathrm{Bun}_G)^\mathrm{temp}$ of tempered objects is compensated by the categories $\mathrm{DMod}(\mathrm{Bun}_M)^\mathrm{temp}$ for all standard Levi subgroups $M \subsetneq G$. This theorem is designed to match exactly with the spectral gluing theorem, an analogous result occurring on the other side of the geometric Langlands conjecture. Along the way, we state and prove several facts that might be of independent interest. For instance, for any parabolic $P \subseteq G$, we show that the functors $\mathrm{CT}_{P,*}:\mathrm{DMod}(\mathrm{Bun}_G) \to \mathrm{DMod}(\mathrm{Bun}_M)$ and $\mathrm{Eis}_{P,*}: \mathrm{DMod}(\mathrm{Bun}_M) \to \mathrm{DMod}(\mathrm{Bun}_G)$ preserve tempered objects, whereas the standard Eisenstein functor $\mathrm{Eis}_{P,!}$ preserves anti-tempered objects.