Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with appendix by Jessica Fintzen)

TL;DR

This paper investigates how the depth of representations and the local Langlands correspondence behave under Weil restriction of groups over local fields, providing a formula to compare depths between the original and Weil-restricted groups.

Contribution

It introduces a depth-comparison formula for Weil-restricted groups, advancing understanding of the local Langlands correspondence in this context.

Findings

Derived a depth-comparison formula for Weil-restricted groups

Analyzed the transition of depth in the local Langlands correspondence

Enhanced understanding of the relationship between G(E) and M(F) representations

Abstract

Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = \mathfrak{R}_{E/F}\, G$ be the reductive group over $F$ obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from $G(E)$ to $M(F)$. We obtain a depth-comparison formula for Weil-restricted groups.