On the local Langlands correspondence. (English) Zbl 1151.11351
Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor (ISBN 7-04-008690-5/3-vol. set). 583-597 (2002).
Author’s summary: “The local Langlands correspondence for \(\text{GL}(n)\) of a non-Archimedean local field \(F\) parametrizes irreducible admissible representations of \(\text{GL}(n,F)\) in terms of representations of the Weil-Deligne group \(\text{WD}_F\) of \(F\). The correspondence, whose existence for \(p\)-adic fields was proved in joint work of the author with R. Taylor [The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Ann. Math. Stud. 151. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)], and then more simply by G. Henniart [Invent. Math. 139, 439–455 (2000; Zbl 1048.11092)], is characterized by its preservation of salient properties of the two classes of representations.
The article reviews the strategies of the two proofs. Both the author’s proof with Taylor and Henniart’s proof are global and rely ultimately on an understanding of the \(\ell\)-adic cohomology of a family of Shimura varieties closely related to \(\text{GL}(n)\). The author’s proof with Taylor provides models of the correspondence in the cohomology of deformation spaces, introduced by Drinfeld, of certain \(p\)-divisible groups with level structure.
The general local Langlands correspondence replaces \(\text{GL}(n,F)\) by an arbitrary reductive group \(G\) over \(F\), whose representations are conjecturally grouped in packets parametrized by homomorphisms from \(\text{WD}_F\) to the Langlands dual group \(^LG\). The article describes partial results in this direction for certain classical groups \(G\), due to Jiang-Soudry and Fargues.
The bulk of the article is devoted to motivating problems that remain open even for \(\text{GL}(n)\). Foremost among them is the search for a purely local proof of the correspondence, especially the relation between the Galois-theoretic parametrization of representations of \(\text{GL}(n,F)\) and the group-theoretic parametrization in terms of Bushnell-Kutzko types. Other open questions include the fine structure of the cohomological realization of the local Langlands correspondence: does the modular local Langlands correspondence of Vigneras admit a cohomological realization?”
An excellent survey highlighting the proofs of and related progress on the local Langlands conjectures!
For the entire collection see [Zbl 0993.00022].
The article reviews the strategies of the two proofs. Both the author’s proof with Taylor and Henniart’s proof are global and rely ultimately on an understanding of the \(\ell\)-adic cohomology of a family of Shimura varieties closely related to \(\text{GL}(n)\). The author’s proof with Taylor provides models of the correspondence in the cohomology of deformation spaces, introduced by Drinfeld, of certain \(p\)-divisible groups with level structure.
The general local Langlands correspondence replaces \(\text{GL}(n,F)\) by an arbitrary reductive group \(G\) over \(F\), whose representations are conjecturally grouped in packets parametrized by homomorphisms from \(\text{WD}_F\) to the Langlands dual group \(^LG\). The article describes partial results in this direction for certain classical groups \(G\), due to Jiang-Soudry and Fargues.
The bulk of the article is devoted to motivating problems that remain open even for \(\text{GL}(n)\). Foremost among them is the search for a purely local proof of the correspondence, especially the relation between the Galois-theoretic parametrization of representations of \(\text{GL}(n,F)\) and the group-theoretic parametrization in terms of Bushnell-Kutzko types. Other open questions include the fine structure of the cohomological realization of the local Langlands correspondence: does the modular local Langlands correspondence of Vigneras admit a cohomological realization?”
An excellent survey highlighting the proofs of and related progress on the local Langlands conjectures!
For the entire collection see [Zbl 0993.00022].
Reviewer: Olaf Ninnemann (Berlin)
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 |
