Explicit Jacquet-Langlands correspondence. II: The case of equal degree and residual characteristic
The local Deligne-Kazhdan correspondence relates square integrable representations of GL(\(n,F\)), \(F\) a \(p\)-adic field, with those on an inner form \(G'\), see [\textit{P. Deligne}, \textit{D. Kazhdan} and \textit{M.-F. Vignéras}, in: Représentations des groupes réductifs sur un corps local, 33--117 (1984; Zbl 0583.22009)], or [\textit{Y. Z. Flicker}, J. Anal. Math. 49, 135--202 (1987; Zbl 0656.10024)], for any \(n\) and any inner form \(G'\), and [\textit{Y. Z. Flicker}, J. Ramanujan Math. Soc. 5, 107--122 (1990; Zbl 0722.11028)], for the special case of anisotropic \(G'\); the case of \(n=2\) is in [\textit{H. Jacquet} and \textit{R. P. Langlands}, Automorphic forms on GL(2) (1970; Zbl 0236.12010)]. The proof is global, using the trace formula and character relations. This paper reproves the local result by purely local means (explicit construction of cuspidal representations using types, à la Carayol, Kutzko, E.-W. Zink and others) when \(n=p\) (then \(G'\) is anisotropic). Part I, cf. \textit{G. Henniart}, Prog. Math. 108, 85--114 (1993; Zbl 0961.11018).
scientific article; zbMATH DE number 1491677