Formules de caractères pour l'induction automorphe

Abstract

Let E/F be a finite cyclic extension of p-adic fields, of degree d, and let κ be a character of F^× with kernel N_E/F (E^×). Automorphic induction corresponds, via the Langlands correspondence, to inducing Galois representations from E to F. To a smooth irreducible representation τ of GL_m(E) automorphic induction attaches a smooth irreducible representation π of GL_md(F) which is equivalent to (κ ○ det) ⊗ π. When π is generic the relation between τ and π is expressed by saying that a certain character function attached to τ is proportional to another character function attached to π and the choice of an intertwining operator A of (κ ○ det) ⊗ π onto π. Here we normalize A through Whittaker models so that the proportionality constant—we prove—does not depend on τ. This is used in current work of C. J. Bushnell and the first author to give an explicit description of the Langlands correspondence for cuspidal smooth irreducible representations of GL_n(F) when n is prime to p. In the present paper we also give a proof of the fundamental lemma for automorphic induction when p is at most n, thus completing J.-L. Waldspurger's result when p > n.