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 ρ 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 • ρ. We study the relation between the Swan exponents Sw(ρ) and Sw(R•ρ), 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 ρ our main results determine Sw(Sym 2 ρ) and Sw(∧ 2 ρ) from Sw(ρ) when the residue characteristic p of F is odd, and bound them in terms of Sw(ρ) when p is 2. In that case where p is 2 we conjecture stronger bounds, for which we provide evidence.