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On the local Langlands correspondence for tori. (English) Zbl 1187.11045

Cunningham, Clifton (ed.) et al., Ottawa lectures on admissible representations of reductive \(p\)-adic groups. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4493-9/hbk). Fields Institute Monographs 26, 177-183 (2009).
This is an exposition of the local Langlands correspondence (LLC) for tori, which first appeared in an unpublished manuscript of Langlands “Representations of algebraic groups, 1968. Mimeographed notes, Yale University”. It is a complement to Mezo’s article on the local Langlands correspondence.
In this article, the author gives an elementary and explicit exposition of this result. The author only concentrates on the local non-archimedean case, and only derives the LLC for tori, not the general cohomological isomorphism. This approach uses only simple results from the cohomology of groups and some basic facts from local class field theory. The author gives a characterization of the LLC for tori, which seems not to be in the literature, and in the end, he proves a depth-preserving theorem when the torus is split over a tamely ramified extension.
Let \(k\) be a non-archimedean local field, and let \(W\) the Weil group of \(k\). Let \(T\) be a torus over \(k\), and let \(T^\vee\) be the dual torus of \(T\), which is defined over a complex field. The main theorem is stated as follows:
There is a unique family of homomorphisms \(\phi_T:Hom(T(k),\mathcal{C}^*)\mapsto H^1(W,T^W)\) with the following properties:
(1) \(\phi_T\) is an additive functorial in \(T\), i.e., it is a morphism between two additive functors from the category of tori over \(k\) to the category of abelian groups;
(2) for \(T=R_{k'/k}(\mathcal{G}_m)\), where \(k'/k\) is a finite Galois extension, \(\phi_T\) is the isomorphism obtained from the local class field theorem and Shapiro’s lemma.
Moreover, \(\phi_T\) is an isomorphism for all \(T\) over \(k\), and the following holds:
(2’) for \(T=R_{k'/k}(\mathcal{G}_m)\), where \(k'/k\) is a finite separable extension, \(\phi_T\) is the isomorphism as in (2).
For the entire collection see [Zbl 1165.22001].

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11S25 Galois cohomology
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