Stable distribution - Knowlpedia

Let $G$ be a connected reductive group over a local field $F$ . An invariant distribution $D$ on $C_c^\infty(G(F))$ is stable when its regular-semisimple part depends only on stable conjugacy. Equivalently, on test functions supported in the strongly regular semisimple locus, $D$ factors through the space of stable orbital integral data rather than depending on the individual $G(F)$ -conjugacy classes inside a stable class. A distribution may also have singular support; its extension across the nonregular locus must satisfy the corresponding stable-germ conditions.

Interpretation

This functional definition remains meaningful when a distribution has no pointwise character function.

Packet characters

For a tempered L-packet $\Pi_\varphi$ on a quasi-split group, the appropriately normalized sum

S\Theta_\varphi=\sum_{\pi\in\Pi_\varphi}\Theta_\pi

is expected, and in many established cases known, to be stable. Other characters of the parameter's component group give generally unstable weighted sums whose transfers come from endoscopic groups.

Global role

The stable trace formula rewrites invariant trace-formula distributions as combinations of stable distributions on endoscopic groups. Stability is therefore the distributional language in which packet transfer and functoriality become visible.

References

Robert P. Langlands and Diana Shelstad, “On the definition of transfer factors,” Mathematische Annalen 278 (1987), 219–271.
James Arthur, “A stable trace formula. I. General expansions,” Journal of the Institute of Mathematics of Jussieu 1 (2002), 175–277. DOI.