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**The local Langlands conjecture for \(GL(n)\) over a \(p\)-adic field, \(n<p\).**
*(English)*
Zbl 0921.11060

Let \(F\) be a \(p\)-adic field and \(n\) a positive integer. Let \({\mathcal A}(n,F)\) denote the set of equivalence classes of irreducible admissible representations of \(GL(n,F)\), and let \({\mathcal G}(n,F)\) denote the set of equivalence classes of \(n\)-dimensional complex representations of the Weil-Deligne group on \(F\) on which Frobenius acts semisimply. The local Langlands conjecture asserts that there exists a unique bijection \(\pi\mapsto \sigma(\pi)\) from \({\mathcal A}(n,F)\) to \({\mathcal G}(n,F)\) satisfying a given set of properties. The author puts the final nail (and a big one) on this important conjecture for the case \(n<p\).

Correspondences \(\pi\mapsto \sigma(\pi)\) in this case were established by G. Henniart [Ann. Sci. Éc. Norm., Supér. (4) 21, 497-544 (1988; Zbl 0666.12013)], C. Bushnell and A. Fröhlich [Gauss sums and \(p\)-adic division algebras, Lect. Notes Math. 987 (1983; Zbl 0507.12008)] and by the author [M. Harris, Invent. Math. 129, 75-119 (1997; Zbl 0886.11029)] using ideas of H. Carayol [Ann. Sci. Éc. Norm. Supér. (4) 19, 409-468 (1986; Zbl 0616.10025)]. The correspondence in M. Harris (loc. cit.) is used here. It was constructed by geometrical methods, works for every \(n\) and \(p\) and is compatible with global correspondences.

Some properties of these local correspondences were established by Moy, by Reimann and Henniart, and by Bushnell, Henniart and Kutzko. The missing piece, which is established here, is the uniqueness of such a correspondence in the case \(n<p\). By a result of Henniart, the uniqueness will follow from an equality of \(\varepsilon \) factors for pairs of corresponding representations. Namely, the author proves the following for the correspondence \(\pi\mapsto \sigma(\pi)\) given in M. Harris (loc. cit.). Let \(\pi_1\) and \(\pi_2\) be cuspidal representations of \(GL_{n_1}(F)\) and \(GL_{n_2}(F)\) respectively, with \(n_1,n_2< p\). Then \(\varepsilon (\pi_1\times \pi_2,s)= \varepsilon (\sigma(\pi_1)\otimes \sigma(\pi_2),s)\) where the left hand side epsilon factor is the one constructed by H. Jacquet, I. Piatetski-Shapiro and J. Shalika [Am. J. Math. 105, 367-464 (1983; Zbl 0525.22018)] and the right hand side epsilon factor is the one constructed by R. Langlands [On the functional equation of Artin’s \(L\)-functions (unpublished manuscript)] and P. Deligne [Modular functions of one variable II, Lect. Notes Math. 349, 501-597 (1973; Zbl 0271.14011)].

The author’s idea for the proof of this equality is to “embed” \(\pi_1\) and \(\pi_2\) into cuspidal automorphic representations of \(GL_{n_1}\) and \(GL_{n_2}\) respectively, and to “embed” \(\sigma(\pi_1)\) and \(\sigma(\pi_2)\) into finite-dimensional complex representations of the absolute Galois group of a certain number field. The embedding is such that the unramified components of the global objects correspond under the local Langlands correspondence. Once such embeddings are achieved, it is possible to obtain the desired epsilon factors equality using a well known game with functional equations and gamma factors employed by Miyake, Casselman, Gelbart, Jacquet, Piateski-Shapiro and Shalika, and Henniart among others.

Hence, the main result of this paper is the global embeddings mentioned above. This is a difficult result. The author employs a clever scheme using some of the deepest results in this field, namely, base change and automorphic induction of Arthur and Clozel and results of Clozel and \(\lambda\)-adic Galois representations associated with automorphic forms. He also develops a theory of non-normal automorphic induction for some very special cases that are required in the proof. In the last part of the paper, the author treats the general local Langlands conjecture and reduces the proof to a certain conjecture which generalizes results of Carayol for \(n=2\). We remark that a proof of this conjecture which establishes the general local Langlands conjecture was recently announced by the author and Richard Taylor. Some details of this proof and an alternate proof of Henniart are provided in a March 1999 Bourbaki talk by Carayol.

Correspondences \(\pi\mapsto \sigma(\pi)\) in this case were established by G. Henniart [Ann. Sci. Éc. Norm., Supér. (4) 21, 497-544 (1988; Zbl 0666.12013)], C. Bushnell and A. Fröhlich [Gauss sums and \(p\)-adic division algebras, Lect. Notes Math. 987 (1983; Zbl 0507.12008)] and by the author [M. Harris, Invent. Math. 129, 75-119 (1997; Zbl 0886.11029)] using ideas of H. Carayol [Ann. Sci. Éc. Norm. Supér. (4) 19, 409-468 (1986; Zbl 0616.10025)]. The correspondence in M. Harris (loc. cit.) is used here. It was constructed by geometrical methods, works for every \(n\) and \(p\) and is compatible with global correspondences.

Some properties of these local correspondences were established by Moy, by Reimann and Henniart, and by Bushnell, Henniart and Kutzko. The missing piece, which is established here, is the uniqueness of such a correspondence in the case \(n<p\). By a result of Henniart, the uniqueness will follow from an equality of \(\varepsilon \) factors for pairs of corresponding representations. Namely, the author proves the following for the correspondence \(\pi\mapsto \sigma(\pi)\) given in M. Harris (loc. cit.). Let \(\pi_1\) and \(\pi_2\) be cuspidal representations of \(GL_{n_1}(F)\) and \(GL_{n_2}(F)\) respectively, with \(n_1,n_2< p\). Then \(\varepsilon (\pi_1\times \pi_2,s)= \varepsilon (\sigma(\pi_1)\otimes \sigma(\pi_2),s)\) where the left hand side epsilon factor is the one constructed by H. Jacquet, I. Piatetski-Shapiro and J. Shalika [Am. J. Math. 105, 367-464 (1983; Zbl 0525.22018)] and the right hand side epsilon factor is the one constructed by R. Langlands [On the functional equation of Artin’s \(L\)-functions (unpublished manuscript)] and P. Deligne [Modular functions of one variable II, Lect. Notes Math. 349, 501-597 (1973; Zbl 0271.14011)].

The author’s idea for the proof of this equality is to “embed” \(\pi_1\) and \(\pi_2\) into cuspidal automorphic representations of \(GL_{n_1}\) and \(GL_{n_2}\) respectively, and to “embed” \(\sigma(\pi_1)\) and \(\sigma(\pi_2)\) into finite-dimensional complex representations of the absolute Galois group of a certain number field. The embedding is such that the unramified components of the global objects correspond under the local Langlands correspondence. Once such embeddings are achieved, it is possible to obtain the desired epsilon factors equality using a well known game with functional equations and gamma factors employed by Miyake, Casselman, Gelbart, Jacquet, Piateski-Shapiro and Shalika, and Henniart among others.

Hence, the main result of this paper is the global embeddings mentioned above. This is a difficult result. The author employs a clever scheme using some of the deepest results in this field, namely, base change and automorphic induction of Arthur and Clozel and results of Clozel and \(\lambda\)-adic Galois representations associated with automorphic forms. He also develops a theory of non-normal automorphic induction for some very special cases that are required in the proof. In the last part of the paper, the author treats the general local Langlands conjecture and reduces the proof to a certain conjecture which generalizes results of Carayol for \(n=2\). We remark that a proof of this conjecture which establishes the general local Langlands conjecture was recently announced by the author and Richard Taylor. Some details of this proof and an alternate proof of Henniart are provided in a March 1999 Bourbaki talk by Carayol.

Reviewer: Ehud Moshe Baruch (Rehovot)

### MSC:

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F80 | Galois representations |

22E50 | Representations of Lie and linear algebraic groups over local fields |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |