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**Proof of the local Langlands conjecture for \(\text{GL}_n\): works of Harris-Taylor and Henniart.
(Preuve de la conjecture de Langlands locale pour \(\text{GL}_n\): Travaux de Harris-Taylor et Henniart.)**
*(French)*
Zbl 0958.11077

Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque 266, 191-243, Exp. No. 857 (2000).

This is an excellent survey of the proofs of the local Langlands conjecture for \(\text{GL}_n\), by M. Harris and R. Taylor [On the geometry and cohomology of some simple Shimura variety. Annals of Mathematics Studies 151. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)] and by G. Henniart [Invent. Math. 139, 439–455 (2000; Zbl 1048.11092)]. It is the non-abelian generalization of local class field theory, and it predicts, for a local non-archimedean field \(K\), the existence of a bijection between the isomorphism classes of irreducible smooth representations of \(\text{GL}_n(K)\) and the isomorphism classes of representations of degree \(n\) of the Weil-Deligne group \(W_K'\), which are \(Fr\)-semisimple. This bijection satisfies a certain list of natural properties (which make it unique), one of which is the preservation of local \(L\) and \(\varepsilon\) factors for pairs.

The proof of Harris and Taylor is geometric in nature. The key ingredient is the study of the reduction at bad places of certain Shimura varieties, which allow to associate Galois representations \(\Sigma\) to certain cusp form \(\Pi\) on \(\text{GL}_n(\mathbb{A}_F)\), where \(F\) is a quadratic imaginary extension of a totally real number field \(F^+\), such that at place \(w\), which splits (as \(\nu \overline{\nu}\)) in \(F\), \(K\) is the completion of \(F^+\) at \(w\). (\(\Pi\) is also such that \(\widehat{\Pi}= \overline{\Pi}\), \(\Pi_\infty\) has a regular infinitesimal character and \(\Pi_\nu\) belongs to a discrete series. \(\Pi\) is taken to be the base change of an automorphic representation \(\pi\) on \(\text{GL}_n (\mathbb A_{F^+})\).) Harris and Taylor show that the restriction of \(\Sigma\) at \(w\) depends only on \(\pi_w\). An important role here is played by the results of Kottwitz and Clozel which associate to \(\Pi\) a compatible system \(\rho_{\lambda,\Pi}\) of \(l\)-adic representations of degree \(n\) of \(\text{Gal} (\overline{F}/F)\), such that at almost all finite places \(\alpha\) of \(F\), \(\Pi_\alpha\) and the restriction of \(\rho_{\lambda,\Pi}\) to the corresponding local Galois group are compatible under the Langlands correspondence (up to a certain twist).

The proof of Henniart establishes the bijection above in the reverse direction. It reduces first to writing the map for irreducible representations \(\sigma\) of \(\text{Gal} (\overline{K}/K)\), with determinant of finite order, and then, using Brauer’s theorem, expressing \(\sigma\), viewed as a virtual representation, as a sum \(\sum n_i \operatorname {Ind}_{L_i}^{K_i} \chi_i\), where \(n_i\) are integers, \(L_i\) is a degree \(d_i\) extension of \(K_i\) and \(\chi_i\) is a character of \(L_i^*\), viewed as a character of \(W_{L_i}\) by local class field theory. Although the extensions \(L_i/K\) are not necessarily Galois extensions, Henniart succeeds, using results of Harris, in performing automorphic induction, and associates to \((L_i, \chi_i)\) a “local automorphic induction representation” \(\pi_i\) of \(\text{GL}_{d_i} (K)\), so that \(\sigma\) corresponds to \(\sum n_i [\pi_i]\), the sum taken in the free \(\mathbb Z\)-module on the isomorphism classes of irreducible, supercuspidal representations of \(\text{GL}_n(K)\), \(m\in \mathbb N\). Here \([\pi_i]\) is the sum of the elements in the cuspidal support of \(\pi_i\).

The survey gives a very good idea of both proofs, philosophy as well as technicalities, all preceded by a good introduction of the basic relevant notions: representations and local factors on the Galois side and on \(\text{GL}_n(K)\) side, explaining how automorphic representations come into play, base change, automorphic induction, and the Jacquet-Langlands correspondence, and finally, a nice treatment of the geometric aspects: non-abelian Lubin-Tate theory and Shimura varieties.

For the entire collection see [Zbl 0939.00019].

The proof of Harris and Taylor is geometric in nature. The key ingredient is the study of the reduction at bad places of certain Shimura varieties, which allow to associate Galois representations \(\Sigma\) to certain cusp form \(\Pi\) on \(\text{GL}_n(\mathbb{A}_F)\), where \(F\) is a quadratic imaginary extension of a totally real number field \(F^+\), such that at place \(w\), which splits (as \(\nu \overline{\nu}\)) in \(F\), \(K\) is the completion of \(F^+\) at \(w\). (\(\Pi\) is also such that \(\widehat{\Pi}= \overline{\Pi}\), \(\Pi_\infty\) has a regular infinitesimal character and \(\Pi_\nu\) belongs to a discrete series. \(\Pi\) is taken to be the base change of an automorphic representation \(\pi\) on \(\text{GL}_n (\mathbb A_{F^+})\).) Harris and Taylor show that the restriction of \(\Sigma\) at \(w\) depends only on \(\pi_w\). An important role here is played by the results of Kottwitz and Clozel which associate to \(\Pi\) a compatible system \(\rho_{\lambda,\Pi}\) of \(l\)-adic representations of degree \(n\) of \(\text{Gal} (\overline{F}/F)\), such that at almost all finite places \(\alpha\) of \(F\), \(\Pi_\alpha\) and the restriction of \(\rho_{\lambda,\Pi}\) to the corresponding local Galois group are compatible under the Langlands correspondence (up to a certain twist).

The proof of Henniart establishes the bijection above in the reverse direction. It reduces first to writing the map for irreducible representations \(\sigma\) of \(\text{Gal} (\overline{K}/K)\), with determinant of finite order, and then, using Brauer’s theorem, expressing \(\sigma\), viewed as a virtual representation, as a sum \(\sum n_i \operatorname {Ind}_{L_i}^{K_i} \chi_i\), where \(n_i\) are integers, \(L_i\) is a degree \(d_i\) extension of \(K_i\) and \(\chi_i\) is a character of \(L_i^*\), viewed as a character of \(W_{L_i}\) by local class field theory. Although the extensions \(L_i/K\) are not necessarily Galois extensions, Henniart succeeds, using results of Harris, in performing automorphic induction, and associates to \((L_i, \chi_i)\) a “local automorphic induction representation” \(\pi_i\) of \(\text{GL}_{d_i} (K)\), so that \(\sigma\) corresponds to \(\sum n_i [\pi_i]\), the sum taken in the free \(\mathbb Z\)-module on the isomorphism classes of irreducible, supercuspidal representations of \(\text{GL}_n(K)\), \(m\in \mathbb N\). Here \([\pi_i]\) is the sum of the elements in the cuspidal support of \(\pi_i\).

The survey gives a very good idea of both proofs, philosophy as well as technicalities, all preceded by a good introduction of the basic relevant notions: representations and local factors on the Galois side and on \(\text{GL}_n(K)\) side, explaining how automorphic representations come into play, base change, automorphic induction, and the Jacquet-Langlands correspondence, and finally, a nice treatment of the geometric aspects: non-abelian Lubin-Tate theory and Shimura varieties.

For the entire collection see [Zbl 0939.00019].

Reviewer: David Soudry (Tel-Aviv)

### MSC:

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

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

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

14L05 | Formal groups, \(p\)-divisible groups |

11G18 | Arithmetic aspects of modular and Shimura varieties |