The unitary dual for the multiplicative group of arbitrary division algebras over local fields

Abstract

Let F be a locally compact, nondiscrete, non-Archimedean field (i.e., a finite extension of Qp or a field of formal Laurent series over a finite field), and let D 2 be a division algebra of degree n over F, so that [D: F] = n . In this paper, we give a construction of all irreducible unitary representations of Dx . There does not seem to be an easy way to parametrize these representations; Theorem 5.5 is a start in that direction. The problem of determining (Dx) ^ is interesting not only in itself, but also for the light it sheds on the problem of determining all irreducible supercuspidal representations of GL, (F) . The problems are connected in two ways. The construction of supercuspidals for GL,(F) has used procedures quite similar to those used to construct (DX) ^. In previous papers, (DX) has been determined when p2 = n or p2 t n. (See [11, 10, 3, 4, 5].) The supercuspidal representations of GL,(F) are known when p t n and when n is the product of at most two distinct primes (see [12, 17, 2, 6]), and there is usually a close similarity in methods of construction to the corresponding case for division algebras. Furthermore, one important method of proving that the construction of supercuspidals is complete uses the Matching Theorem of Deligne and Kazhdan (see [8]), which sets up a natural correspondence between (Dx) ^ and the set of discrete series representations of GL, (F) . (See, e.g., [2] or [ 17] for this kind of proof.) It seems likely that the construction in this paper will also apply to give a construction of all supercuspidal representations of GL, (F). This matter will be dealt with in a future paper. (In this connection, it is worth noting the results in [15] about supercuspidals in GL, (F), n the product of two primes.) Describing the general method of the construction requires some notation. Let k be the residue class field of F, and suppose that k has q elements. Recall (from, e.g., Chapter I of [18]) that D contains an unramified extension