Twisted Conjugacy Classes

Twisted conjugacy is conjugation by a group element that is also acted upon by an automorphism. It includes \(\sigma\)-conjugacy (for Frobenius / Galois actions) and stable conjugacy (for cohomologically- adjusted equivalence) as special cases.

Definition

Let \(G\) be a group with an automorphism \(\sigma : G \to G\). Two elements \(x, y \in G\) are \(\sigma\)-conjugate (or \(\sigma\)-twisted conjugate) if there exists \(g \in G\) with \[ y \;=\; g \, x \, \sigma(g)^{-1}. \]

The equivalence classes are the \(\sigma\)-conjugacy classes, denoted \(B(G, \sigma)\) (or just \(B(G)\) when \(\sigma\) is canonical).

When \(\sigma = \mathrm{id}\), this reduces to ordinary conjugacy.

Geometric interpretation

For an algebraic group \(G\) over \(k\) with automorphism \(\sigma\) (e.g.\ Frobenius for a group over \(\bar{\mathbb{F}_q}\) coming from a group over \(\mathbb{F}_q\)): \(\sigma\)-conjugacy classes parametrise twisted forms.

Specifically: a 1-cocycle \(c \in Z^1(\langle \sigma \rangle, G)\) is the same as an element of \(G\); the cohomology class is the \(\sigma\)-conjugacy class.

Examples

Frobenius twisted conjugacy

For \(G(\bar{\mathbb{F}_q})\) with Frobenius \(F\): by Lang's theorem, every \(F\)-conjugacy class contains the identity (\(F\)-conjugacy is trivial). This is why finite reductive groups are studied via the Lang-Steinberg map \(g \mapsto g^{-1} F(g)\), which is surjective.

Kottwitz setting

For \(G(\breve F)\) with \(\sigma\) the Frobenius of the unramified extension: \(B(G) = G(\breve F) / \sigma\)-conjugacy is the Kottwitz set, the central object of \(p\)-adic geometric Langlands.

Stable conjugacy

In the theory of stable conjugacy, one declares \(\gamma, \gamma' \in G(k)\) stably conjugate if they are \(G(\bar k)\)-conjugate. The stable conjugacy class \([\gamma]_{\mathrm{st}}\) is a Galois-twisted version of the conjugacy class, classified by: \[ \mathrm{Stab}(\gamma) \backslash H^1(k, Z_G(\gamma)), \] where \(Z_G(\gamma)\) is the centraliser.

For a regular semisimple \(\gamma\): \(Z_G(\gamma)\) is a maximal torus, and the classification is purely \(H^1\)-cohomological.

Galois-cohomological reformulation

\(\sigma\)-conjugacy classes in \(G\) for an order-\(n\) automorphism \(\sigma\) correspond to: \[ B(G, \sigma) \;\cong\; H^1(\mathbb{Z}/n, G). \]

More generally, for a profinite group \(\Gamma\) acting on \(G\): 1-cocycles in \(H^1(\Gamma, G)\) are essentially \(\sigma\)-conjugacy classes (with \(\sigma\) ranging over \(\Gamma\)).

Counting and orbit structure

For finite groups (or \(G(\mathbb{F}_q)\) for finite reductive groups):

Theorem (Burnside, Lang). The number of \(\sigma\)-conjugacy classes in \(G(\bar{\mathbb{F}_q})\) for \(\sigma = \mathrm{Frob}\) equals the number of \(\mathrm{Frob}\)-fixed conjugacy classes in \(G(\bar{\mathbb{F}_q})\) — equivalently, the number of conjugacy classes in \(G(\mathbb{F}_q)\).

This is the basis of Deligne-Lusztig theory: representations of finite reductive groups indexed by Frobenius-fixed conjugacy classes of pairs \((T, \theta)\) with \(T\) a maximal torus and \(\theta\) a character.

Stable vs.\ ordinary conjugacy

Over a field \(k\), elements \(\gamma, \gamma' \in G(k)\):

Conjugate if they are \(G(k)\)-conjugate.
Stably conjugate if they are \(G(\bar k)\)-conjugate.

The "stable" notion is coarser; the obstruction to lifting stable to ordinary conjugacy lies in \(H^1(k, Z_G(\gamma))\).

For regular semisimple \(\gamma\) over a local field: stable conjugacy classes split into multiple ordinary classes, parametrised by characters of a finite cohomology group — the endoscopic component group.

Role in Langlands

Kottwitz set \(B(G)\): \(\sigma\)-conjugacy classes index inner forms and isocrystals.
Stable trace formula: integrates over stable conjugacy classes, not just conjugacy classes; the difference is captured by the endoscopic transfer factors.
Twisted endoscopy (Kottwitz-Shelstad): generalisation to twisted conjugacy with non-trivial Galois action; foundation of base change trace formula identities.