Deligne constructed a remarkable local system on P 1 − {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 -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 ramification for any reductive group.

Abstract. We define the Heegner-Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between (1) The r-th central derivative of the quadratic base change L-function associated to an everywhere unramified cuspidal automorphic representation π of PGL 2 ; (2) The self-intersection number of the π-isotypic component of the Heegner-Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross-Zagier formula for higher derivatives of L-functions.

Abstract. For any Kac-Moody group G with Borel B, we give a monoidal equivalence between the derived category of B-equivariant mixed complexes on the flag variety G/B and (a certain completion of) the derived category of B ∨ -monodromic mixed complexes on the enhanced flag variety G ∨ /U ∨ , here G ∨ is the Langlands dual of G. We also prove variants of this equivalence, one of which is the equivalence between the derived category of U -equivariant mixed complexes on the partial flag variety G/P and certain "Whittaker model" category of mixed complexes on G ∨ /B ∨ . In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in [BGS96].

We provide geometric constructions of modules over the graded Cherednik algebra H gr ν and the rational Cherednik algebra H rat ν attached to a simple algebraic group G together with a pinned automorphism θ. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit C * -actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When θ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules Lν (triv) of H gr ν and of H rat ν . We give a formula for the dimension of Lν (triv) and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details. Part 3. Representations 49 7. Geometric modules of the graded Cherednik algebra 49 7.1. The H gr -action on the cohomology of homogeneous affine Springer fibers 49 7.2. The polynomial representation of H gr 51 7.3. The global sheaf-theoretic action of H gr 52 7.4. Local-global compatibility 54 8. Geometric modules of the rational Cherednik algebra 55 8.1. The polynomial representation of H rat 55 8.2. The H rat -action on the cohomology of homogeneous affine Springer fibers 56 8.3. The perverse filtration 57 8.4. The global sheaf-theoretic action of H rat 60 8.5. Proof of Theorem 8.2.3(1) 61 8.6. Frobenius algebra structure and proof of Theorem 8.2.3(2) 62 8.7. Langlands duality and Fourier transform 65 9.Examples 66 9.1. Algebraic theory 67 9.2. Type 2 A 2 , m 1 = 2 67 9.3. Type C 2 , m = 2 68 9.4. Type 2 A 3 , m 1 = 2 69 9.5. Type 2 A 4 , m 1 = 2 71 9.6. Type G 2 , m = 3 74 9.7. Type G 2 , m = 2 75 9.8. Type 3 D 4 , m 1 = 6 76 9.9. Type 3 D 4 , m 1 = 3 77 9.10. Dimensions of L ν (triv): tables and conjectures 78 Appendix A. Dimension of affine Springer fibers for quasi-split groups 78 Appendix B. Codimension estimate on the Hitchin base 80 References 82Let ν ∈ Q. The Galois action of Z(1) on t ν ∈ F ∞ gives a character which we denote by ζ → ζ ν . This character only depends on the class of ν in Q/Z. Concretely, if ν = a/b in lowest terms with b > 0, then the corresponding character is Z(1)