Depth preserving property of the local Langlands correspondence for non-quasi-split unitary groups

TL;DR

This paper extends the depth preserving property of the local Langlands correspondence from quasi-split to non-quasi-split unitary groups by leveraging the local theta correspondence and related splittings.

Contribution

It generalizes the depth preservation result to non-quasi-split cases using the local theta correspondence and compares different splittings for metaplectic covers.

Findings

Depth preservation holds for non-quasi-split unitary groups.

Comparison of splittings enhances understanding of the local theta correspondence.

The approach unifies different constructions of splittings in the context of unitary groups.

Abstract

In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups (arXiv:1804.10901) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence established by Gan--Ichino. To combine them, we compare splittings for metaplectic covers of unitary groups constructed by Kudla with those constructed by Pan.