Note on the non-preservation of depth

TL;DR

This paper demonstrates that in the local Langlands correspondence for tori over local fields of characteristic p, the property of depth is not necessarily preserved, providing explicit counterexamples.

Contribution

It constructs explicit examples showing that depth preservation fails in the local Langlands correspondence for tori in characteristic p.

Findings

Depth is not preserved in certain cases.

Counterexamples illustrate the failure of depth preservation.

Highlights limitations of the local Langlands correspondence for tori.

Abstract

Let $K$ be a local field of characteristic $p$. We consider the local Langlands correspondence for tori, and construct examples for which depth is not preserved.