2010/05/16 by Jochen Heinloth, Heinloth, Jochen, Ngô Bảo Châu +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1005.2765
openalex publication_date 2010/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Deligne constructed a remarkable local system on \bP1-\0,∞\ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct l-adic Kloosterman sheaves for any reductive group in a uniform way, and describe the local and global monodromy of these Kloosterman sheaves. In particular, they give motivic Galois representations with exceptional monodromy groups G2,F4,E7 and E8. This also gives an example of the geometric Langlands correspondence with wild ramifications for any reductive group.