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On the local Langlands conjecture for \(\mathrm{GL}(n)\): the cyclic case. (English) Zbl 0588.12010

Let \(F\) be a non-Archimedean local field. Let \(G\circ (n)\) denote the set of equivalence classes of irreducible representations of degree n of the Weil group \(W_ F\) and \(A\circ (n)\) the set of equivalence classes of irreducible supercuspidal representations of \(\mathrm{GL}(n,F)\). According to the Langlands conjectures there should exist a sequence of bijections \(\pi_n: G\circ (n)\to A\circ (n)\), preserving \(L\)- and \(\varepsilon\)-factors (for pairs) and extending class field theory. Let \(G^c(n)\) be the subset of \(G\circ (n)\) consisting of the classes of representations which are induced from a cyclic extension of degree \(n\) of \(F\) and let \(A^c(n)\) be the subset of \(A\circ (n)\) consisting of \(\pi\) such that \(\pi \otimes \chi =\pi\) for some character \(\chi\) of \(F^{\times}\) of order \(n\).
In the paper a natural bijection \(\pi^c_n: G^c(n)\to A^ c(n)\) is constructed, which, among other properties, preserves \(L\)- and \(\varepsilon\)-factors. A similar construction had been given by Kazhdan, without establishing the property with respect to \(L\)- and \(\varepsilon\)-factors, however. Moreover, in the author’s work the proofs are complete also in positive characteristic.
The proof uses global means, such as the trace formula for the multiplicative group \(A^{\times}\) of a central division algebra \(A\) of degree \(n\) over a global field and the correspondence between automorphic representations for \(\mathrm{GL}(n)\) and for \(A^{\times}.\)
The result of the paper can be applied to get a bijection \(G\circ (p)\to A\circ (p)\), where \(p = \) residue characteristic of \(F\).

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
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