The Wavefront Conjecture

Arthur packets are collections of representations of reductive groups that are predicted by Arthur’s conjectures to share specific spectral properties. Each representation has a wavefront set — a geometric invariant describing the singular support of its character distribution. Jiang conjectured that the wavefront set of any representation in an Arthur packet is bounded above by the unipotent orbit attached to the packet’s Arthur parameter.

Hazeltine, Liu, and Lo prove the conjecture for representations in local Arthur packets of classical groups, under a standard hypothesis on the local Langlands correspondence. The proof proceeds through the endoscopic structure of the packets — the way each representation contributes to the trace formula through endoscopic transfer.

The wavefront set measures the “size” of the representation in a precise geometric sense: larger wavefront sets correspond to representations that are more singular, more spread out in the unitary dual. The conjecture says Arthur’s parameter — a homomorphism from a specific group to the Langlands dual — controls how singular the representations in its packet can be.

The bound is sharp: the packet always contains at least one representation that achieves the maximum wavefront set predicted by the parameter. The bound is also uniform: every representation in the packet is bounded by the same orbit.

A geometric bound on an analytic invariant, controlled by an algebraic parameter. The Langlands program predicting representation theory through number theory, confirmed for one more class of cases.