Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured P 1 C having differential Galois group G2, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pullbacks we construct families of G2-connections which are not of these types.

We propose a new method to construct rigid G-automorphic representations and rigid $${\widehat{G}}$$ G ^ -local systems for reductive groups G. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.

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We propose a new method to construct rigid G-automorphic representations and rigid G-local systems for reductive groups G. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties. Contents 1. Introduction 1 2. Euphotic representations 5 3. Euphotic automorphic data 7 4. Hecke eigencategory and local systems 11 5. An example in type G 2 17 6. The hyperspecial cases 19 7. Detailed analysis of stabilizers 23 8. Potential Examples 36 Appendix A. Factorizable module categories 41 References 47