Classicality of derived Emerton--Gee stack II: generalised reductive groups

TL;DR

The paper addresses extending the Emerton–Gee stack to general and generalized reductive groups by employing a Tannakian framework to define the stack $\mathcal X_G$ and constructing a derived counterpart $\mathfrak X_G$ on the transversal absolute prismatic site. It proves $\mathcal X_G$ is a formal algebraic stack locally of finite presentation, and that the underlying classical stack of $\mathfrak X_G$ coincides with $\mathcal X_G$, with classicality established for connected reductive and generalized reductive groups (via a modified derived stack and, where applicable, the Langlands dual context). The work connects derived Laurent $F$-crystals with $G$-structure to derived Galois representations, using Herr complexes and deformation theory to build obstruction theories and pro-cotangent comparisons (GR17), and leverages the PQ24 framework to handle generalised reductive groups including $L$-groups. Collectively, these results provide a robust geometric and derived-analytic framework for local Langlands parameters across a broad class of groups, enabling a unified spectral-side perspective and potential advances toward categorical $p$-adic Langlands correspondences.

Abstract

We use the Tannakian formalism to define the Emerton--Gee stack for general groups. For a flat algebraic group G over Z_p, we are able to prove the associated Emerton--Gee stack is a formal algebraic stack locally of finite presentation over Spf(Z_p). We also define a derived stack of Laurent F-crystals with G-structure on the absolute prismatic site, whose underlying classical stack is proved to be equivalent to the Emerton--Gee stack. In the case of connected reductive groups, we show that the derived stack of Laurent F-crystals with G-structure is classical in the sense that when restricted to truncated animated rings, it is the étale sheafification of the left Kan extension of the Emerton--Gee stack along the inclusion from classical commutative rings to animated rings. Moreover, when G is a generalised reductive group, the classicality result still holds for a modified version of the Emerton--Gee stack. In particular, this completes the picture that the derived stack of local Langlands parameters for the Langlands dual group of a reductive group is classical.

Theorems & Definitions (145)

• Definition 1.1
• Theorem 1.2: Theorem \ref{['main-1']}
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Definition 1.6
• Theorem 1.7: Theorem \ref{['classical']}
• Theorem 1.8: Theorem \ref{['main-connected']}
• Remark 1.9
• Definition 1.10: PQ24