Twisted conjugacy - Knowlpedia

Definition

Let $G$ be a group and let $\theta$ be an automorphism of $G$ . Two elements $x,y\in G$ are $\theta$ -conjugate if

y=gx\theta(g)^{-1}

for some $g\in G$ . Equivalently, they lie in the same orbit for the twisted action $g\mathbin{\cdot_\theta}x=gx\theta(g)^{-1}$ . When $\theta$ is the identity this is ordinary conjugacy.

The twisted centralizer of $x$ is the stabilizer $G_{x,\theta}=\{g:gx\theta(g)^{-1}=x\}$ .

Frobenius and sigma-conjugacy

If $G$ is defined over a finite field or local field and $\sigma$ is Frobenius, then twisted conjugacy is usually called $\sigma$ -conjugacy. The Kottwitz set $B(G)$ is the set of $\sigma$ -conjugacy classes in $G(\breve F)$ .

Twisted conjugacy also appears in the nonidentity components of semidirect products such as $\widehat G\rtimes\langle\theta\rangle$ : ordinary conjugacy within the coset of the dual group $\widehat G\theta$ becomes $\theta$ -conjugacy in $\widehat G$ .

Stable variant

For algebraic groups one can likewise compare twisted conjugacy over the base field with twisted conjugacy over an algebraic closure. The latter gives stable twisted classes, whose rational orbits are controlled by Galois Galois cohomology of the twisted centralizer.

References

Robert E. Kottwitz, “Isocrystals with additional structure,” Compositio Mathematica 56 (1985), 201–220. Numdam.
Robert E. Kottwitz, “Stable trace formula: cuspidal tempered terms,” Duke Mathematical Journal 51 (1984), 611–650.