Rad HAZU, Matematičke znanosti, Vol. 28 (2024), 151-184.

ON SWAN EXPONENTS OF SYMMETRIC AND EXTERIOR SQUARE GALOIS REPRESENTATIONS

Guy Henniart and Masao Oi

Department of Mathematics (Hakubi center), Kyoto University, Kitashirakawa, Oiwake-cho, Sakyo-ku, Kyoto, Japan
e-mail: masaooi@math.kyoto-u.ac.jp

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(Sym²ρ) and Sw(∧²ρ) 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.

2020 Mathematics Subject Classification.   11F80, 11S37, 22E50.

Key words and phrases.   Swan exponents, Galois representations, local Langlands correspondence.

DOI: https://doi.org/10.21857/yvjrdcde6y

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