##
**Local tame lifting for \(\text{GL}(n)\). II: Wildly ramified supercuspidals.**
*(English)*
Zbl 0920.11079

Astérisque. 254. Paris: Société Mathématique de France, iv, 105 pp. (1999).

[For Part I see Publ. Math., Inst. Hautes Étud. Sci. 83, 105–233 (1996; Zbl 0878.11042).]

Let \(F\) be a nonarchimedean local field. The local Langlands conjectures demand the existence of a canonical bijection between the set \({\mathcal A}_n(F)\) of irreducible supercuspidal representations of \(\text{GL}_n(F)\) and the set \({\mathcal G}_n(F)\) of irreducible \(n\)-dimensional complex representations of the Weil group of \(F\). In the case \(n=1\) this is given by the isomorphism of class field theory. Recently M. Harris and R. Taylor have announced a solution of this problem in giving such a bijection having the postulated properties. Their approach is geometric in nature and depends on global constructions. It is thus not clear how their map interacts with the classification of supercuspidals given in [C. Bushnell and P. Kutzko, The admissible dual of \(\text{GL}(N)\) via compact open subgroups. Annals of Mathematics Studies. 129. Princeton, NJ: Princeton University Press (1993; Zbl 0787.22016)]. The present book is intended as a step towards an explicit description of the Langlands correspondence in this crucial case.

In the book two things are done: There is given an extension of base change and an explicit correspondence map. Both is done under the following restrictions: One only considers the case \(n=p^m\), where \(p\) is the residue characteristic of \(F\) and one only considers the sets of wildly ramified representations \({\mathcal A}^{wr}_{ p^m} (F)\), \({\mathcal G}^{wr}_{p^m}(F)\). (On the Galois side wildly ramified representations are those which stay irreducible when restricted to the wild inertia group.) The central theme of the book is a lifting map extending base change: \[ l_{K/F}:{\mathcal A}^{wr}_{p^m} (F)\to{\mathcal A}^{wr}_{p^m}(K) \] for any tame, possibly non-Galois, extension \(K/F\). Having this lifting at hand the correspondence \({\mathcal A}^{wr}_{p^m}(F)\to{\mathcal G}^{wr}_{p^m}(F)\) is easily given: At first one considers representations which are gotten by successive induction from characters in which case the correspondence is nailed down by its functorial properties and the fact that it extends class field theory in the \(\text{GL}_1\)-case. Next, the authors use the fact that any wildly ramified representation is related via extended base change to a representation of this type. The second part only works in characteristic zero, whereas the construction of the lifting map \(l_{K/F}\) is valid in any characteristic.

Let \(F\) be a nonarchimedean local field. The local Langlands conjectures demand the existence of a canonical bijection between the set \({\mathcal A}_n(F)\) of irreducible supercuspidal representations of \(\text{GL}_n(F)\) and the set \({\mathcal G}_n(F)\) of irreducible \(n\)-dimensional complex representations of the Weil group of \(F\). In the case \(n=1\) this is given by the isomorphism of class field theory. Recently M. Harris and R. Taylor have announced a solution of this problem in giving such a bijection having the postulated properties. Their approach is geometric in nature and depends on global constructions. It is thus not clear how their map interacts with the classification of supercuspidals given in [C. Bushnell and P. Kutzko, The admissible dual of \(\text{GL}(N)\) via compact open subgroups. Annals of Mathematics Studies. 129. Princeton, NJ: Princeton University Press (1993; Zbl 0787.22016)]. The present book is intended as a step towards an explicit description of the Langlands correspondence in this crucial case.

In the book two things are done: There is given an extension of base change and an explicit correspondence map. Both is done under the following restrictions: One only considers the case \(n=p^m\), where \(p\) is the residue characteristic of \(F\) and one only considers the sets of wildly ramified representations \({\mathcal A}^{wr}_{ p^m} (F)\), \({\mathcal G}^{wr}_{p^m}(F)\). (On the Galois side wildly ramified representations are those which stay irreducible when restricted to the wild inertia group.) The central theme of the book is a lifting map extending base change: \[ l_{K/F}:{\mathcal A}^{wr}_{p^m} (F)\to{\mathcal A}^{wr}_{p^m}(K) \] for any tame, possibly non-Galois, extension \(K/F\). Having this lifting at hand the correspondence \({\mathcal A}^{wr}_{p^m}(F)\to{\mathcal G}^{wr}_{p^m}(F)\) is easily given: At first one considers representations which are gotten by successive induction from characters in which case the correspondence is nailed down by its functorial properties and the fact that it extends class field theory in the \(\text{GL}_1\)-case. Next, the authors use the fact that any wildly ramified representation is related via extended base change to a representation of this type. The second part only works in characteristic zero, whereas the construction of the lifting map \(l_{K/F}\) is valid in any characteristic.

Reviewer: Anton Deitmar (Heidelberg)

### MSC:

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

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

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

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

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |