Euphotic representations and rigid automorphic data

Abstract

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

Rigidity method for automorphic forms over function fields

“…3.3.4. Example (Euphotic automorphic data, [20]). This is a further generalization of the epipelagic automorphic data that weakens the assumption that ϕ ∈ V * P is stable.…”

“…Such automorphic datum (K S , K S ) is not guaranteed to be weakly rigid. In [20], for P ∞ = G(O ∞ ), we give a complete list of pairs (ϕ, Q) (where ϕ ∈ g * is semisimple, and Q ⊂ G is a parabolic subgroup) satisfying the condition (3.2). In [20] we prove that in these cases, for a generic choice of the Kummer sheaf K χ on B ϕ , (K S , K S ) is weakly rigid, but not always geometrically rigid.…”