Davenport-Hasse relations and an explicit Langlands correspondence. (English) Zbl 1029.11063
Summary: Let \(F\) be a finite extension of \(\mathbb{Q}_p\), and \({\mathcal W}_F\) the Weil group of \(F\) relative to some algebraic closure of \(\mathbb{Q}_p\). An irreducible representation of \({\mathcal W}_F\) is wildly ramified if its dimension is a power of \(p\) and it is not equivalent to an unramified twist of itself. Let \({\mathcal G}_m^{\text{wr}} (F)\) be the set of equivalence classes of such representations of dimension \(p^m\). Let \({\mathcal A}_m^{\text{wr}} (F)\) be the set of equivalence classes of irreducible supercuspidal representations \(\pi\) of \(\text{GL}_{p^m} (F)\) such that \(\pi\) is not a nontrivial unramified twist of itself. The Langlands conjecture predicts the existence of a canonical bijection \({\mathcal G}_m^{\text{wr}} (F)\rightarrow{\mathcal A}_m^{\text{wr}} (F)\) for each \(m\). Two such bijections are known. The first, denoted \({\mathcal L}_m\), is a special case of the construction of a Langlands correspondence due to M. Harris and R. Taylor [The geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton Univ. Press (2001; Zbl 1036.11027)]. The other, denoted \(\pi_m\), is due to the authors [C. J. Bushnell and G. Henniart, Local tame lifting for \(\text{GL}(N)\). II: Wildly ramified supercuspidals, Astérisque 254 (1999; Zbl 0920.11079)]; this construction is quite explicit, and totally different from that of Harris and Taylor. This paper investigates the relation between the two. It is shown that, for \(\sigma\in {\mathcal G}_m^{\text{wr}} (F)\), the representations \(\pi(\sigma)\), \({\mathcal L}(\sigma)\) differ by an unramified twist of order dividing \(p^m\). The authors’ construction is based on a certain tame lifting map. If this satisfies a certain “Hasse-Davenport relation” on local constants of pairs, then \(\pi_m= {\mathcal L}_m\).
MSC:
| 11S37 | Langlands-Weil conjectures, nonabelian class field theory |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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