Kloosterman sheaves for reductive groups


Deligne constructed a remarkable local system on $\mathbb{P}^1-\{0,\infty\}$ 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 $\ell$-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 $G_2,F_4,E_7$ and $E_8$. This also gives an example of the geometric Langlands correspondence with wild ramification for any reductive group.


Jochen Heinloth

University of Amsterdam, Korteweg-de Vries Institute of Mathematics, 1098 XH Amsterdam, The Netherlands

Bao-Chaû Ngô

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637

Zhiwei Yun

Department of Mathematics, Stanford University, Stanford, CA 94305