Colin J. Bushnell, Guy Henniart
Local tame lifting for GL(n) III: explicit base change and Jacquet-Langlands correspondence
Let F be a finite extension of ?p with p ?2, and D a central F-division algebra of dimension p2m. Let ? be an irreducible supercuspidal representation of GLpm(F). The Jacquet-Langlands correspondence associates to ? an irreducible smooth representation ?D of Dx, determined up to isomorphism by a character relation. Using a variant of the description of irreducible supercuspidal representations of GLn(F) as induced representations, due to Bushnell and Kutzko, along with a parallel description for Dx due to Broussous, we give an explicit realization of the correspondence ? ? ?D in the case where ? is totally ramified. This is a step towards our main result. Let K?F be a finite unramified extension, and ? a totally ramified supercuspidal representation of GLpm(F). Base change, in the sense of Arthur and Clozel, gives a totally ramified supercuspidal representation bK?F ? of GLpm(K). In earlier work, the authors gave an explicit definition of a representation ?K?F ? and showed that ?K?F ? = bK?F ? when p does not divide the degree of K?F. We complete this by showing that ?K?F ? = bK?F ? for all K?F. The proof relies on evaluating the twisted character of ?K?F ? in terms of the character of ?D and then using the explicit Jacquet-Langlands correspondence. Many of the central arguments remain valid when F is a non-Archimedean local field of odd positive characteristic.
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2005, 03/2005
Pages: 39 - 100
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