Explicit Geometric Langlands Correspondence for Rigid Local Systems

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Probably the best known example of a rigid local system is the solution sheaf of the Gaussian hypergeometric equation on the complex-projective line P^1 punctured at 0, 1 and infinity. In this context, the rigidity condition says that this solution sheaf (which is a local system) is determined up to isomorphism by the local monodromy around the punctures. This local monodromy at a point is given as a linear relation between a fundamental solution of the differential equation and its analytic continuation along a simple path around the missing point. The Gaussian equation has the special property that all its singular points are regular singular. Roughly speaking this means that the solutions are subject to some growth condition. Rigid local systems arising from regular singular differential equations in this way have found application in inverse Galois theory and the construction of motives with exceptional motivic Galois groups. More generally one can define a similar notion for differential equations resp. connections on trivial vector bundles which are not necessarily regular singular. In this case such a connection is called rigid if its isomorphism class is determined uniquely by local formal data. There are essentially two ways to construct rigid connections resp. in the setting of a finite base field, l-adic local systems. The first of these is a theorem of Katz & Arinkin which states that any irreducible rigid connection on an open subset of P^1 can be constructed from a connection of rank one by iterating Fourier-Laplace transform, twisting with a connection of rank one or coordinate changes by a Möbius transform. In the regular singular (resp. tamely ramified) case, one can replace Fourier-Laplace transform by middle convolution defined by Katz. This method was used by Dettweiler & Reiter to classify tamely ramified rigid local systems with monodromy group the simple exceptional algebraic group of type G_2. Additionally I used this method to construct new rigid irregular connections with differential Galois group of type G_2. The second way of constructing rigid local systems is the following. Heinloth, Ngô & Yun use the geometric Langlands correspondence to construct local systems as eigensystems of Hecke-eigensheaves on the moduli space of G-bundles with level structure. They constructed Kloosterman sheaves for reductive groups, realizing several exceptional algebraic groups as geometric monodromy groups.The aim of this project is the generalization of Heinloth, Ngô & Yun’s construction to obtain new classes of rigid local systems. Additionally we wish to reobtain certain known examples. In particular we hope to obtain an automorphic interpretation of Katz’s hypergeometric sheaves. These are l-adic analogues and generalizations of the Gaussian hypergeometric equation mentioned above.