On Swan exponents of symmetric and exterior square Galois representations

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.