Stable Conjugacy Classes

Motivation

Over a non-algebraically-closed field $F$, a single geometric conjugacy class in $G(\bar F)$ may split into several $G(F)$-rational conjugacy classes. The trace formula and the theory of endoscopy require a notion that groups these together: the stable conjugacy class.

Definition (semisimple case)

Two semisimple elements $\gamma, \gamma' \in G(F)$ are stably conjugate if they are conjugate over $\bar F$: $$\gamma' = g \gamma g^{-1} \quad \text{for some } g \in G(\bar F).$$

This is a coarser equivalence relation than $G(F)$-conjugacy. The stable conjugacy class of $\gamma$ is the set of all $G(F)$-conjugacy classes inside its geometric conjugacy class.

Cohomological parametrisation

The set of $G(F)$-conjugacy classes within the stable class of $\gamma$ is parametrised by the Galois cohomology set $$\ker\bigl(H^1(F, C_G(\gamma)) \to H^1(F, G)\bigr),$$ where $C_G(\gamma)$ is the centraliser of $\gamma$ in $G$.

For $G = \mathrm{GL}_n$, all $H^1$ vanish, so stable $=$ rational conjugacy. The difference first appears for groups with non-trivial $H^1$.

Regular semisimple case

For regular semisimple $\gamma$ with connected centraliser $T = C_G(\gamma)$ (a maximal torus), the $G(F)$-classes within the stable class of $\gamma$ are parametrised by $$\ker\bigl(H^1(F, T) \to H^1(F, G)\bigr) \cong H^1(F, T)/\mathrm{im}.$$ This finite set measures the failure of Hilbert 90 for the torus $T$.

Role in endoscopy

Endoscopic groups $H$ of $G$ are defined so that:

• Stable conjugacy classes of $H(F)$ map to stable conjugacy classes of $G(F)$.
• The transfer of orbital integrals from $G$ to $H$ (or vice versa) respects stable conjugacy classes.

The fundamental lemma (Ngô, 2010) and the transfer conjecture (Langlands–Shelstad) are stated in terms of stable orbital integrals: $$SO_\gamma^G(f) = \sum_{\delta \leftrightarrow \gamma} \Delta(\gamma, \delta)\, O_\delta^H(f^H),$$ where the sum is over conjugacy classes $\delta$ within the stable class of $\gamma$.

Unipotent case

For unipotent elements over $p$-adic fields, stable conjugacy is more subtle. Two unipotent elements are stably conjugate if they lie in the same orbit over $\bar F$ and correspond under the Springer correspondence to the same local system. The parametrisation involves Lusztig's theory of character sheaves.