Microlocal Arthur packets as a quasi-canonical basis of stable distributions

Source abstract

We show that microlocal Arthur packets can be reconstructed from an otherwise underdetermined system of linear equations arising from Kashiwara's local index theorem. The key is a geometric rigidity property of Arthur orbits which, together with Fourier symmetry, uniquely determines the microlocal multiplicities defining these packets. This realizes the associated Arthur distribution as the unique normalized element in the intersection of two filtered spaces of standard stable distributions related by Fourier transform. Under standard enhanced local Langlands hypotheses and Aubert compatibility, this gives an alternative representation-theoretic definition of microlocal Arthur packets for general connected reductive $p$-adic groups that does not use twisted endoscopy and ensures stability through the stability of tempered packet distributions. In the unipotent setting, we use this characterization to prove agreement between endoscopic and microlocal Arthur packets for split $\mathrm{SO}_{2n+1}$ when the residue characteristic is at least $6n+4$. We further use this characterization to compute all unipotent Arthur packets with unramified real infinitesimal character for inner-to-split adjoint exceptional $p$-adic groups. In this exceptional setting, we prove the microlocal weak Arthur packet conjecture and give counterexamples to the microlocal analogues of the least- and greatest-parameter conjectures.