B(G)

For a connected reductive group \(G\) over a non-archimedean local field \(F\) (or equivalently over the Witt-vector ring), the Kottwitz set \(B(G)\) is the set of \(\sigma\)-conjugacy classes in \(G(\breve F)\), where \(\breve F\) is the completion of the maximal unramified extension of \(F\) and \(\sigma\) is the Frobenius. \(B(G)\) is the basic arithmetic invariant of \(G\) controlling its forms, isocrystals, and Newton stratification.

Definition

Let \(F\) be a non-archimedean local field with residue field \(\mathbb{F}_q\), completion of the maximal unramified extension \(\breve F = \widehat{F^{\mathrm{unr}}}\), Frobenius automorphism \(\sigma : \breve F \to \breve F\). For a connected reductive group \(G\) over \(F\):

\[ B(G) \;=\; G(\breve F) \;/\; \sigma\text{-conjugacy}, \]

where two elements \(g, h \in G(\breve F)\) are \(\sigma\)-conjugate if \(h = x g \sigma(x)^{-1}\) for some \(x \in G(\breve F)\).

The Kottwitz invariants

Kottwitz introduced two fundamental invariants \(\kappa : B(G) \to X^*(Z(\hat G))^{\Gamma}\) and \(\nu : B(G) \to (X_*(T)_\mathbb{Q})^{\Gamma}/W\) (the Newton homomorphism):

\(\kappa\) is the Kottwitz homomorphism, classifying the image in the algebraic fundamental group;
\(\nu\) is the Newton point (an element of the rational cocharacter chamber).

Kottwitz's theorem: the pair \((\kappa, \nu)\) is injective on \(B(G)\).

Examples

\(G = \mathrm{GL}_n\)

By Dieudonné–Manin, \(\sigma\)-conjugacy classes in \(\mathrm{GL}_n(\breve F)\) are classified by isocrystals of dimension \(n\), which are parametrised by their Newton polygons (= multisets of \(n\) rational numbers with denominators \(\le n\)).

Equivalently, \(B(\mathrm{GL}_n) \cong\) {Newton polygons in \([0, 1]\) of slope length \(\le n\)}.

\(G\) a torus

For a torus \(T\), \(B(T) = X_*(T)^{\Gamma}\) (Galois invariants of the cocharacter lattice). The Newton map is the identity composed with the embedding into \((X_*(T)_\mathbb{Q})^{\Gamma}\).

General \(G\)

\(B(G)\) is partitioned into Newton strata indexed by \(\nu \in (X_*(T)_\mathbb{Q})^{\Gamma}/W\), with each stratum a torsor for the central kernel. The set has a partial order (the Bruhat-like order on Newton points), with basic strata corresponding to elements where \(\nu\) is central.

Why B(G) matters

Affine Deligne-Lusztig varieties. Indexed by elements of \(B(G)\) paired with elements of an affine Weyl group; their geometry encodes much of the \(p\)-adic representation theory.
Local Langlands. The set \(B(G)\) parametrises the "inner forms" of \(G\) showing up in Kottwitz's parametrisation of local L-packets.
Shimura varieties. The reduction modulo \(p\) of Shimura varieties has a Newton stratification indexed by elements of \(B(G)\) (Rapoport).
Hodge–Newton decomposition. Compatibility of Newton and Hodge filtrations on a \(\mathrm{GL}_n\) isocrystal.

References

Kottwitz, Isocrystals with additional structure I, II (Compos. Math. 1985, 1997).
Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996).
Görtz, Affine Deligne–Lusztig varieties and the geometry of Shimura varieties (notes, 2009).