Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

TL;DR

This paper establishes a detailed correspondence between Arthur packets of real unitary groups and $p$-adic symplectic or orthogonal groups, enabling transfer of results and computations between these settings.

Contribution

It introduces an explicit relation between real and $p$-adic Arthur packets, connecting translation and Jacquet functors, and constructs a correspondence of Langlands parameter stacks.

Findings

Explicit correspondence of Arthur packets between real and $p$-adic groups.

Relation between Zuckerman's translation functor and the Jacquet functor.

Connection of Kazhdan-Lusztig polynomials and microlocal geometry across settings.

Abstract

We establish an explicit correspondence of certain Arthur packets between real unitary groups and $p$-adic symplectic or orthogonal groups. This allows one to compute Arthur packets of real unitary groups by translating results from the $p$-adic side. A main ingredient in our proof is an explicit relation between Zuckerman's translation functor on the real side and the Jacquet functor on the $p$-adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and $p$-adic groups. Our approach also allows one to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and $p$-adic sides.