Sigma-Conjugacy Classes

The notion of \(\sigma\)-conjugacy generalises ordinary conjugacy: it depends on a group action by an automorphism \(\sigma\), most often the Frobenius of an unramified extension of a local field. It is the fundamental equivalence relation underlying the Kottwitz set \(B(G)\).

Definition

Let \(G\) be a group equipped with an automorphism \(\sigma\) (e.g.\ \(G = G(L)\) for a reductive group \(G\) over a field \(F\) with a field automorphism \(\sigma\) of an overfield \(L/F\)). Two elements \(g, h \in G\) are \(\sigma\)-conjugate if there exists \(x \in G\) such that \[ h \;=\; x \, g \, \sigma(x)^{-1}. \]

The equivalence classes are called \(\sigma\)-conjugacy classes, and the set of classes is denoted \(B(G, \sigma)\) (or simply \(B(G)\) when the choice of \(\sigma\) is canonical).

When \(\sigma = \mathrm{id}\), \(\sigma\)-conjugacy is ordinary conjugacy.

The Lang–Steinberg setting

Let \(G\) be a connected algebraic group over a finite field \(\mathbb{F}_q\) with geometric Frobenius \(\sigma = \mathrm{Frob}_q\).

Lang–Steinberg theorem. The map \[ G(\bar{\mathbb{F}_q}) \;\to\; G(\bar{\mathbb{F}_q}), \quad g \mapsto g^{-1} \sigma(g) \] is surjective. Hence every \(\sigma\)-conjugacy class contains the identity (modulo cohomology), and \(H^1(\sigma, G(\bar{\mathbb{F}_q})) = 1\).

This is the foundational vanishing result that makes Galois cohomology manageable over finite fields.

The Kottwitz setting

Let \(F\) be a non-archimedean local field with residue field \(\mathbb{F}_q\), and let \(\breve F\) be the completion of the maximal unramified extension \(F^{\mathrm{unr}}\). The Frobenius \(\sigma\) acts on \(\breve F\) (and on \(G(\breve F)\)) as the \(q\)-th power map on residues, lifted to \(\breve F\).

The set \[ B(G) \;=\; G(\breve F) / \sigma\text{-conjugacy} \] is the Kottwitz set of \(G\), the central object of B(G).

In contrast to the Lang–Steinberg setting, the Kottwitz set is typically non-trivial because \(\breve F\) (unlike \(\bar{\mathbb{F}_q}\)) is not "Brauer-trivial" enough — there are non-split inner forms, isocrystals with non-trivial Newton polygon, etc.

Why \(\sigma\)-conjugacy and not ordinary conjugacy

Many arithmetic invariants (isocrystals, inner forms, Galois twists) are naturally classified by \(\sigma\)-conjugacy rather than conjugacy:

An \(F\)-rational form of an object over \(\bar F\) corresponds to a \(\sigma\)-conjugacy class of cocycles \(\mathrm{Gal}(\bar F/F) \to \mathrm{Aut}(X)\); for cyclic Galois groups this reduces to \(\sigma\)-conjugacy of a single cocycle value.
An isocrystal of dimension \(n\) over \(\breve F\) is a pair \((V, F)\) with \(V \cong \breve F^n\) and \(F : V \to V\) a \(\sigma\)-linear isomorphism; isomorphism classes are \(\sigma\)-conjugacy classes in \(\mathrm{GL}_n(\breve F)\).

Examples

\(G = \mathrm{GL}_1 = \mathbb{G}_m\)

\(G(\breve F)^\times = \breve F^\times\); \(\sigma\)-conjugacy is the equivalence \(b \sim a \cdot b \cdot \sigma(a)^{-1}\). Because \(\breve F^\times\) is \(\sigma\)-divisible enough (Hilbert 90 unramified), \(\sigma\)-conjugacy classes are parametrised by \(\mathbb{Z} = \breve F^\times / (\text{unramified norms})\) (the valuation).

So \(B(\mathbb{G}_m) = \mathbb{Z}\), with class of \(b\) determined by \(v(b)\).

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

\(B(\mathrm{GL}_n)\) = isocrystals up to iso = Newton polygons (Dieudonné-Manin).

\(G\) a torus

For a torus \(T\) over \(F\), \(B(T) = X_*(T)^\Gamma\) (Galois invariants of the cocharacter lattice). The bijection sends a cocharacter \(\mu : \mathbb{G}_m \to T\) to the class of \(\mu(\pi) \in T(\breve F)\), where \(\pi\) is a uniformiser.

Maps and invariants

\(\sigma\)-conjugacy classes carry two basic functors:

Newton map \(\nu : B(G) \to (X_*(T)_{\mathbb{Q}})^\Gamma / W\) — extracts the slope filtration (for \(\mathrm{GL}_n\), the Newton polygon).
Kottwitz map \(\kappa : B(G) \to \pi_1(G)_\Gamma\) — projects to the algebraic fundamental group modulo Frobenius coinvariants.

Together \((\nu, \kappa)\) injects \(B(G)\) into a product of combinatorial sets ([[node:kottwitz.b_of_g|Kottwitz theorem]]).

References

Kottwitz, Isocrystals with additional structure I, II (Compositio Math. 1985, 1997).
Steinberg, Endomorphisms of linear algebraic groups (Mem. AMS 1968).
Borel, Linear Algebraic Groups (1991), §16 (Lang–Steinberg).