Representations of tamely ramified p-adic division and matrix algebras

Abstract

Let F be a p-adic number field, and let D be a central division algebra of index n over F. We assume, throughout, that n is not divisible by p. W, denotes the Weil group of F. In [6], Corwin and Howe parametrized the set of irreducible representations of D*. Koch and Zink took up the subject in [ 131 and described a bijection between this set and the set of irreducible representations, of dimension dividing n, of W,. In [ 121, Howe constructed a similar map from the set of n-dimensional irreducible representations of W, into the set of supercuspidal representations of Gl,(F). Using the results of Bernstein-Zelevinski [2, 191 and DeligneeKazhdan [ 1, 161, Moy proved (cf. [ 151) that this map is a bijection, too. But the way he presents the correspondence, he has to exclude the case p = 2. Both correspondences are not canonical in the sense of Langlands’ philosophy (cf. [ 143). In [3], Bushnell and Friihlich showed that, in the division algebra case, this defect can be eliminated by certain modifications (twist with tame characters). In [15], Moy tried to construct a similar modification of the correspondence in the matrix algebra case. Unfortunatley, his construction is based on c-factor calculations, which are incorrect in some cases. To explain this, we use the notations of [ 151. In the proof of Theorem 3.544, Moy claims that his Lemma 3.5.36 implies the equation tir = ti’. But if E is a quadratic extension of F and c is not a square in E*, X+ and K’ differ by a nontrivial quadratic character. This can be seen using the following general fact.