Tame multiplicity and conductor for local Galois representations

TL;DR

This paper establishes a bound on the multiplicity of characters in irreducible Galois representations of local fields, linking it to the Swan exponent, and clarifies the structure of the representation's restriction to inertia groups.

Contribution

It proves that the multiplicity of any character of a specific cyclic group in an irreducible Galois representation is bounded by its Swan exponent, answering a question posed by Mark Reeder.

Findings

Bound on character multiplicity in Galois representations

Structure of the representation's restriction to inertia groups

Connection between Swan exponent and representation multiplicities

Abstract

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $\sigma$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(\sigma)$ the Swan exponent of $\sigma$. Assume $\sw(\sigma) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $\sigma(\Cal I_F) = \sigma(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\sigma$ of any character of $\varSigma$ is bounded by $\sw(\sigma)$.