On Swan exponents of symmetric and exterior square Galois representations

TL;DR

This paper investigates how Swan exponents of Galois representations change under symmetric and exterior square operations, with implications for the local Langlands correspondence and explicit Langlands parameter descriptions.

Contribution

It provides explicit formulas and bounds for Swan exponents of symmetric and exterior square representations, especially distinguishing cases where the residue characteristic is 2 or odd.

Findings

Explicit determination of Swan exponents for odd residue characteristic

Bounds on Swan exponents when residue characteristic is 2

Applications to the local Langlands correspondence for classical groups

Abstract

Let $F$ be a local non-Archimedean field and $E$ a finite Galois extension of $F$, with Galois group $G$. If $\rho$ is a representation of $G$ on a complex vector space $V$, we may compose it with any tensor operation $R$ on $V$, and get another representation $R\circ\rho$. We study the relation between the Swan exponents $\mathrm{Sw}(\rho)$ and $\mathrm{Sw}(R\circ\rho)$, with a particular attention to the cases where $R$ is symmetric square or exterior square. Indeed those cases intervene in the local Langlands correspondence for split classical groups over $F$, via the formal degree conjecture, and we present some applications of our work to the explicit description of the Langlands parameter of simple cuspidal representations. For irreducible $\rho$ our main results determine $\mathrm{Sw}(\mathrm{Sym}^{2}\rho)$ and $\mathrm{Sw}(\wedge^{2}\rho)$ from $\mathrm{Sw}(\rho)$ when the residue characteristic $p$ of $F$ is odd, and bound them in terms of $\mathrm{Sw}(\rho)$ when $p$ is $2$. In that case where $p$ is $2$ we conjecture stronger bounds, for which we provide evidence.