Kottwitz set B(G) - Knowlpedia

Let $E$ be a nonarchimedean local field, let $\breve E$ be the completion of its maximal unramified extension, and let $\sigma$ be Frobenius. For a connected reductive $E$ -group $G$ , the Kottwitz set

B(G)=G(\breve E)/{\sim_\sigma}

is the set of $\sigma$ -conjugacy classes, where

b'\sim_\sigma b \quad\Longleftrightarrow\quad b'=g\,b\,\sigma(g)^{-1}

for some $g\in G(\breve E)$ .

It is a pointed set, not generally a group.

Newton and Kottwitz invariants

A class $[b]$ has two principal invariants:

\nu_b\in \left(X_*(T)_\mathbb Q^+\right)^\Gamma, \qquad \kappa_G(b)\in\pi_1(G)_\Gamma.

Here $X_*(T)$ is the cocharacter lattice of a maximal torus and $\Gamma$ is the relevant Galois group. The Newton point records slopes, while the Kottwitz invariant is the generalized degree. Their compatible pair determines $[b]$ ; more precisely, the map $(\nu,\kappa)$ is injective with a characterized image.

For $G=\operatorname{GL}_n$ , $B(G)$ is the set of isomorphism classes of rank- $n$ isocrystals, $\nu_b$ is the Newton-slope polygon, and $\kappa_G(b)$ is its endpoint.

Basic classes

A class is basic when $\nu_b$ is central. The associated group

G_b(R)= \{g\in G(R\otimes_E\breve E): g\,b=b\,\sigma(g)\}

is then an inner form of $G$ . Basic classes correspond to semistable $G$ -bundles on the Fargues–Fontaine curve.

Bundle classification

The Fargues construction assigns to $b$ a $G$ -bundle $\mathcal E_b$ . On geometric points this gives a bijection

B(G)\xrightarrow{\sim}|\operatorname{Bun}_G|.

The Newton partial order becomes the specialization order among bundle strata.

References

Robert E. Kottwitz, “Isocrystals with additional structure II,” Compositio Mathematica 109 (1997), 255–339. Numdam.
Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” §§I.4 and III.2. arXiv.