Abstract
Let F be a non-Archimedean local field, let n >= 1 be an integer and G = GL(n)(F). Let G' be an inner form of G, so that G' is isomorphic to GL(m) (D), for a central F-division algebra of dimension d(2), md = n. Using the structure theory of Secherre and Stevens, we define a concept of parametric degree for irreducible cuspidal representations of G'. We show that the image, under the Jacquet-Langlands correspondence, of the set of equivalence classes of irreducible cuspidal representations of G is the set of equivalence classes of irreducible cuspidal representations of G' of parametric degree n. In earlier papers, we defined a notion of essential tameness for irreducible cuspidal representations of G. We generalize this to representations of G', and show that it is preserved by the Jacquet-Langlands correspondence. As in the earlier papers, their reducible, essentially tame, cuspidal representations of G admit an explicit parametrization in terms of admissible pairs, which can be explicitly related to the local Langlands correspondence. Here, we generalize this construction to the case of irreducible, essentially tame, cuspidal representations of G' of parametric degree n. We determine completely the behaviour of the two parametrizations relative to the Jacquet-Langlands correspondence. As a consequence, we prove a conjecture of Bushnell and Frohlich of 1983.
Original language | English |
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Pages (from-to) | 469 - 538 |
Number of pages | 70 |
Journal | Pure And Applied Mathematics Quarterly |
Volume | 7 |
Issue number | 3 |
Publication status | Published - Jul 2011 |