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 = GLn(F), and let D be a central F-division algebra of dimension n2. The Jacquet-Langlands correspondence gives a canonical bijection π↦πD from the set of equivalence classes of irreducible, smooth, essentially square-integrable representations π 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 dimτ, for an irreducible smooth representation τ of D× to be of the form πD, for an irreducible supercuspidal representation π 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×.