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Local tame lifting for GL\((n)\). III: Explicit base change and Jacquet-Langlands correspondence. (English) Zbl 1074.11063
Let \(F\) be a finite extension of \({\mathbb Q}_p\), \(p\neq 2\), and \(K/F\) a finite unramified extension. Let \({\mathcal A}_m^{\text{ wr}}(F)\) denote the set of equivalence classes of irreducible, totally unramified, supercuspidal representations of the group \(\text{ GL}_{p^m}(F)\). In earlier work [(I) Publ. Math., Inst. Hautes Étud. Sci. 83, 105–233 (1996; Zbl 0878.11042), (II) Astérisque. 254. Paris: Société Mathématique de France (1999; Zbl 0920.11079)] the authors have constructed the tame lifting map \({\mathcal l}_{K/F}\colon {\mathcal A}_m^{\text{ wr}}(F)\to{\mathcal A}_m^{\text{ wr}}(K)\) in explicit terms using the classification of representations of \(\text{ GL}_{p^m}\) in [C. J. Bushnell, P. C. Kutzko, The admissible dual of \(\text{ GL}(N)\) via compact open subgroups. Annals of Mathematics Studies, 129. Princeton University Press (Princeton, NJ) (1993; Zbl 0787.22016)].
In the present paper they finish the proof begun earlier that the tame lifting coincides with the Langlands base change map, i.e., they give an explicit description of base change. As an intermediate step they give an explicit description of the Jacquet-Langlands correspondence that relates representations in \({\mathcal A}_m^{\text{ wr}}(F)\) to representations of a central simple division algebra over \(F\) of degree \(p^m\).
For Part IV, see Proc. Lond. Math. Soc., III. Ser. 87, No. 2, 337–362 (2003; Zbl 1037.22032).

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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