On the Local Langlands Conjecture in Prime Dimension

Abstract

Let F be a local field of residual characteristic p. Then it is a conjecture of Langlands [JL] that there should be a natural bijection between the set of n-dimensional semisimple representations of the absolute Weil-Deligne group of F and the set of irreducible admissible representations of GL ,(F). Before describing this bijection, we note that by results of Bernstein and Zelevinsky [Z] one may restrict one's attention to irreducible representations of the WeilDeligne group on the one hand and irreducible supercuspidal representations of GL ,(F) on the other hand. In this context, the precise form of the conjecture is that there exists a bijection a -g(a) of the set a12(F) of equivalence classes of continuous, irreducible n-dimensional complex representations of the absolute Weil group, WF of F with the set 10(GLJ( F)) of equivalence classes of admissible irreducible supercuspidal representations of GL ,(F). This bijection satisfies the following conditions: