On the depth of the adjoint representations

TL;DR

This paper establishes an explicit formula for the slope of the adjoint of a Carayol Galois representation, linking it to the depth of representations in the Local Langlands Correspondence for GL_n over non-archimedean fields.

Contribution

It provides a new explicit formula for the slope of the adjoint of a Carayol Galois representation, enhancing understanding of the local Langlands correspondence invariants.

Findings

Explicit formula for the slope of the adjoint of Carayol representations

Clarifies the relationship between Galois slopes and representation depth

Advances the understanding of the local Langlands correspondence invariants

Abstract

Let $F$ be a non-archimedean local field of odd residual characteristic $p$. The depth of a smooth representation of ${\rm GL}_n(F)$ is an invariant of Local Langlands Correspondence (LLC). The analogous notion on the Galois side of LLC is known as the slope of a local Galois representation. The slope is well related to the Swan conductor for irreducible Galois representations, whereas its behavior is subtle for reducible Galois representations. In this article, we provide an explicit formula for the slope of the adjoint of a Carayol representation of the local Galois group.