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

2013/03/22 by Mitya Boyarchenko, Boyarchenko, Mitya, Jared Weinstein +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.1303.5795

arxiv created 2013/03/22 · openalex publication_date 2013/03/22 · arxiv updated 2013/03/26 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

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 π of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, π corresponds to an irreducible representation ρ 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 π and ρ, which is simpler than the existing algebraic approaches (in particular, the use of the Weil representation over finite fields is eliminated).

Related

• On the local Langlands correspondence
• Explicit local Jacquet-Langlands correspondence: the non-dyadic wild case
• Geometric approach to the local Jacquet-Langlands correspondence
• Geometric approach to the explicit local Langlands correspondence
• The Local Langlands Correspondence for Simple Supercuspidal Representations of GL_n(F)