Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

The local Langlands correspondence for GL(n) of a non-Archimedean local field F parametrizes irreducible admissible representations of GL(n, F ) in terms of representations of the Weil-Deligne group WDF of F . The correspondence, whose existence for p-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of s...

a nontrivial additive character of k. We are given a polynomial f = f(x1, ..., xn) in n ≥ 1 variables over k of degree d ≥ 1 which is a “Deligne polynomial”, i.e., its degree d is prime to p and its highest degree term, say fd, is a homogeneous form of degree d in n variables which is nonzero, and whose vanishing, if n ≥ 2, defines a smooth hypersurface in the projective space Pn−1. For a Delig...

Let G be a reductive algebraic group over the local field k. The local Langlands conjecture predicts that the irreducible complex representations π of the locally compact group G(k) can be parametrized by objects of an arithmetic nature: homomorphisms φ from the Weil-Deligne group of k to the complex L-group of G, together with an irreducible representation ρ of the component group of the centr...