Endoscopic transfer and the wavefront upper bound conjecture

TL;DR

This paper verifies a local analogue of Jiang's wavefront set conjecture for certain p-adic group representations, confirming related upper bound conjectures under specific conditions using endoscopic transfer and wavefront computations.

Contribution

It establishes the local wavefront set upper bound for Arthur type representations of split classical p-adic groups, extending previous conjectures with new verification methods.

Findings

Verification of the local wavefront set upper bound conjecture.

Confirmation of related upper bound conjectures under certain conditions.

Application of endoscopic transfer and wavefront set computations in the proof.

Abstract

We verify the local analogue of Jiang's conjecture for the upper bound of the geometric wavefront sets of Arthur type representations of split classical $p$-adic groups with $p\gg 0$, under a certain condition. As a consequence, we also obtain the upper bound conjecture of Kim and the second author, and Hazeltine--Liu--Lo--Shahidi, under the same assumptions. The proof uses Waldspurger's work on the endoscopic transfer supplemented by results of Konno and Varma, as well as the wavefront set computations in the unipotent case by Mason-Brown--Okada and the second author.