Ramification of weak Arthur packets for p-adic groups

TL;DR

This paper explicitly decomposes weak Arthur packets for split odd orthogonal and symplectic p-adic groups into Arthur packets via endoscopic transfer, linking their structure to ramification and geometric properties of the dual group.

Contribution

It provides a detailed description of the decomposition of weak Arthur packets into Arthur packets for specific p-adic groups, introducing weak sphericity and connecting it to geometric and representation-theoretic structures.

Findings

Explicit decomposition of weak Arthur packets into Arthur packets.

Introduction of weak sphericity and its characterization of packets.

Connection between geometric properties of the dual group and packet structure.

Abstract

Weak Arthur packets have long been instrumental in the study of the unitary dual and automorphic spectrum of reductive Lie groups, and were recently introduced in the p-adic setting by Ciubotaru - Mason-Brown - Okada. For split odd orthogonal and symplectic p-adic groups, we explicitly determine the decomposition of weak Arthur packets into Arthur packets that arise from endoscopic transfer. We establish a characterization of the Arthur packets that partake in such decompositions by means of ramification properties of their constituents. A notion of weak sphericity for an irreducible representation is introduced: The property of containing fixed vectors with respect to a (not necessarily hyperspecial) maximal compact subgroup. We show that this property determines the weak Arthur packets in a precise sense. As steps towards this description, we explore alignments between Langlands-type reciprocities for finite and p-adic groups, and their dependence on the geometry of the unipotent locus of the dual Langlands group. Weak sphericity is shown to match with Lusztig's canonical quotient spaces that feature in the geometric theory for Weyl group representations, while the fine composition of weak Arthur packets is found to be governed by the partition of the unipotent locus into special pieces.