Stable trace formula

The stable trace formula is the endoscopic reorganization of the invariant Arthur trace formula into distributions that depend only on stable conjugacy and stable packet data.

Schematically,

I^G(f) = \sum_{\mathfrak e} \iota(G,\mathfrak e)\, \widehat S^{H_{\mathfrak e}}(f^{\mathfrak e}),

where $\mathfrak e$ ranges over elliptic endoscopic data, $f^{\mathfrak e}$ is an endoscopic transfer of $f$ , and $\widehat S^{H_{\mathfrak e}}$ is a stable distribution for the endoscopic group. The actual formula includes Levi, central, and measure data suppressed here.

Why stabilization is needed

The invariant trace formula is invariant under conjugating a test function, but its individual geometric terms can distinguish rational conjugacy classes inside one stable class. On the spectral side, individual characters inside an $L$ -packet are likewise unstable.

Fourier analysis on the cohomological obstruction groups separates these unstable terms into $\kappa$ -pieces. Endoscopic transfer identifies each piece with a stable distribution on an endoscopic group.

Inputs

Stabilization uses:

stable orbital integrals;
endoscopic data;
normalized transfer factors;
local and global transfer;
the fundamental lemma and its weighted variants.

Status and scope

Arthur developed the stabilization of the general invariant trace formula, initially conditional on fundamental lemmas that are now theorems in the required standard settings. Specialized stable formulas may impose quasi-splitness, central-character, test-function, or characteristic hypotheses. “The stable trace formula” names this framework and its precise instances, not one hypothesis-free finite sum.

Consequences

Stable comparison makes packet characters visible and underlies Arthur's classification for symplectic and orthogonal groups, Mok's classification for quasi-split unitary groups, and many cases of automorphic transfer.

References

James Arthur, “A stable trace formula III: proof of the main theorems,” Annals of Mathematics 158 (2003), 769–873. Clay copy.
James Arthur, “An introduction to the trace formula,” §§27–29. Clay Mathematics Proceedings.