Explicit unramified base change: GL( p) of a p-adic field

Abstract

Let F/ Q p be a p-adic local field, and let K/ F be a finite unramified field extension. We consider the class of totally ramified, irreducible supercuspidal representations of GL p ( F). All such representations can be obtained by induction from quasicharacters of open, compact modulo centre subgroups of GL p ( F). This description, due to Kutzko and Moy, suggests an explicit definition of a “lifting” operation which maps such a representation π F to a totally ramified, irreducible supercuspidal representation π K of GL p ( K). We show that, apart from a minor adjustment when p=2, the operation π F ↦ π K coincides with base change in the sense of Arthur and Clozel. This is achieved by calculating directly with the Shintani character relation which defines base change. It relies on identifying certain values of the twisted character of π K with values of the character of a representation of a division algebra over F and then using an explicit description of the Jacquet–Langlands correspondence.