Kloosterman sheaves for reductive groups

By J. Heinloth, Ngo Bao Chau, Zhiwei Yun. 2010

81 citations recorded by the source corpus.

Deligne constructed a remarkable local system on P 1 f 0;1g 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 nd 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 ‘-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 ramication for any reductive group.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/529477866e584e66a88967c8fcfd73fc035930be

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Geometric Satake Correspondence for Ramified Groups (2011, background): “– As is shown [14], Aut⊗( C , ω) can be upgraded into a fppf sheaf on the category of affine schemes over E, and as fppf sheaves Aut⊗( C , ω) ' Aut(H).”
Epipelagic $$L$$L-packets and rectifying characters (2012, background): “[21] construct very interesting l-adic local systems on Gm , which generalize the Kloosterman sheaves introduced by Deligne.”
Algebraic trace functions over the primes (2012, background): “This paper has benefited from discussions and comments from Jean Bourgain, Satadal Ganguly, Paul Nelson, Richard Pink, Ngô Bao Châu, Peter Sarnak, Igor Shparlinski, Akshay Venkatesh, and Zhiwei Yun and it is a pleasure to thank them for their input.”
Epipelagic representations and invariant theory (2013, background): “In this context, recall that a linear functional λ ∈ V̌x,r is semistable if its orbit under Gx does not contain zero in its closure under the Zariski topology on V̌x,r and λ is stable if its orbit is closed and its stabilizer in Gx is finite.”