Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences

Abstract

Let F be a non-Archimedean local field and let E be an unramified extension of F of degree n>1. To each sufficiently generic multiplicative character of E (the details are explained in the body of the paper) one can associate an irreducible n-dimensional representation of the Weil group W_F of F, which corresponds to an irreducible supercuspidal representation \pi\ of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, \pi\ corresponds to an irreducible representation \rho\ of the multiplicative group of the central division algebra over F with invariant 1/n. In this note we give a new geometric construction of the representations \pi\ and \rho, which is simpler than the existing algebraic approaches (in particular, the use of the Weil representation over finite fields is eliminated).