G-bundle on the Fargues-Fontaine curve

Let $G$ be a connected reductive group over a nonarchimedean local field $E$ . A $G$ -bundle on the Fargues–Fontaine curve is a principal $G$ -bundle on $X_{S,E}$ ; equivalently, in Tannakian form, it is an exact tensor functor

\operatorname{Rep}_E(G) \longrightarrow \operatorname{Bun}(X_{S,E}).

As the perfectoid space $S$ varies, these bundles form a $v$ -stack $\operatorname{Bun}_G$ .

Bundle attached to an isocrystal

For $b\in G(\breve E)$ , every representation $\rho:G\to\operatorname{GL}(V)$ gives the isocrystal $(V\otimes_E\breve E,\rho(b)\sigma)$ , hence a vector bundle on the curve. The compatible family of these vector bundles defines a $G$ -bundle $\mathcal E_b$ . Its isomorphism class depends only on $[b]\in B(G)$ .

Classification and strata

On geometric points, the assignment

[b]\longmapsto\mathcal E_b

is a bijection from the Kottwitz set $B(G)$ to $|\operatorname{Bun}_G|$ . The Newton and Kottwitz invariants give a Harder–Narasimhan stratification

\operatorname{Bun}_G = \bigsqcup_{b\in B(G)}\operatorname{Bun}_G^b.

The stratum is a geometric stack, not merely a point; its automorphism geometry contains the group $G_b(E)$ .

Semistability

The bundle $\mathcal E_b$ is semistable exactly when $b$ is basic, meaning that its Newton point is central. On the basic stratum, $G_b$ is an inner form of $G$ . The trivial class gives the open stratum

\operatorname{Bun}_G^1\simeq [*/G(E)].

This embeds the category of smooth $G(E)$ -representations into sheaves on $\operatorname{Bun}_G$ .

Local Langlands role

Hecke modifications of $G$ -bundles at untilt divisors implement the geometric Satake action. The Fargues conjecture and Fargues–Scholze theory seek to organize sheaves on $\operatorname{Bun}_G$ by local $L$ -parameters.

References

Laurent Fargues, “ $G$ -torseurs en théorie de Hodge $p$ -adique,” Compositio Mathematica 156 (2020), 2076–2110. DOI.
Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” Chapter III. arXiv.