Local Jacquet-Langlands correspondence and parametric degrees

Let F be a non-Archimedean local field with finite residue field. Let n be a positive integer, let G = GL_n(F), and let D be a central F-division algebra of dimension n². The Jacquet-Langlands correspondence gives a canonical bijection ^D from the set of equivalence classes of irreducible, smooth, essentially square-integrable representations of G to the set of equivalence classes of irreducible smooth representations of D^!!times;. We give a necessary and sufficient condition, in terms of dim, for an irreducible smooth representation of D^× to be of the form ^D, for an irreducible supercuspidal representation of G, thereby solving an old problem. This relies on the explicit classification of the irreducible smooth representations of G and the parallel classification of the irreducible representations of D^×.This paper was written while the first-named author was visiting, and partly supported by, Université de Paris-Sud. At that time, the second-named author was enjoying the hospitality of the IHES, during a stay at the CNRS granted by Université de Paris-Sud; he would like to thank all those institutions. The work was also partially supported by EU network Arithmetical Algebraic Geometry.Mathematics Subject Classification (2000): 22E50