Stable conjugacy - Knowlpedia

Let $G$ be a connected reductive group over a field $F$ . Two strongly regular semisimple elements $\gamma,\gamma'\in G(F)$ are stably conjugate if they are conjugate over an algebraic closure: there is $g\in G(\overline F)$ such that

\gamma'=g\gamma g^{-1}.

For more general semisimple elements, the standard definition additionally requires the cocycle $g^{-1}\sigma(g)$ to lie in the identity component of the centralizer $G_\gamma$ for every $\sigma$ in the absolute Galois group $\operatorname{Gal}(\overline F/F)$ .

Rational classes inside a stable class

Fix strongly regular $\gamma$ and write $T=G_\gamma$ . The $G(F)$ -conjugacy classes in its stable class are parametrized by a kernel in nonabelian Galois cohomology:

\ker\!\left[ H^1(F,T)\longrightarrow H^1(F,G) \right].

Thus stable conjugacy is coarser than rational conjugacy. Over a local field, the displayed kernel is finite.

For $\operatorname{GL}_n$ , two regular semisimple elements that are conjugate over $\overline F$ are already conjugate over $F$ , so a stable class contains one rational class. Other groups can have several.

Stable invariants

For a split group, the adjoint quotient $G\to T/W$ records a characteristic-polynomial-like invariant. On the strongly regular locus, two elements have the same adjoint-quotient value exactly when they are stably conjugate.

Role in endoscopy

Ordinary orbital integrals distinguish rational conjugacy classes. Endoscopy reorganizes their combinations into stable distributions, beginning with stable orbital integrals. Matching stable classes on an endoscopic group and on $G$ is a prerequisite for defining the transfer factor.

References

Robert E. Kottwitz, “Stable trace formula: elliptic singular terms,” Mathematische Annalen 275 (1986), 365–399. DOI.
Ngô Bảo Châu, “Survey on the fundamental lemma,” §2.1. PDF.