An explicit local geometric Langlands for supercuspidal representations: the toral case

2025/06/16 by Yi, Lingfei
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2506.13179

Abstract

We formulate a conjecture on local geometric Langlands for supercuspidal representations using Yu's data and Feigin-Frenkel isomorphism. We refine our conjecture for a large family of regular supercuspidal representations defined by Kaletha, and then confirm the conjecture for toral supercuspidal representations of Adler whose Langlands parameters turn out to be exactly all the irreducible isoclinic connections. As an application, we establish the conjectural correspondence between global Airy connections for reductive groups and the family of Hecke eigensheaves constructed by Jakob-Kamgarpour-Yi.

Citations

• Geometric Langlands for Irregular Theta Connections and Epipelagic Representations
• Stokes phenomenon of Kloosterman and Airy connections
• Differential Galois groups of G-connections with Coxeter singularities
• Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers
• Hypergeometric sheaves for classical groups via geometric Langlands
• Types for tame p-adic groups
• Irreducible connections admit generic oper structures
• Geometric representations of graded and rational Cherednik algebras
• Epipelagic representations and rigid local systems
• Frenkel-Gross' irregular connection and Heinloth-Ngô-Yun's are the same
• Epipelagic L-packets and rectifying characters
• Simple wild L-packets
• Kloosterman sheaves for reductive groups
• Any flat bundle on a punctured disc has an oper structure
• D-modules on the affine flag variety and representations of affine Kac-Moody algebras
• Local Geometric Langlands Correspondence: the Spherical Case
• Distinguished tame supercuspidal representations
• Supercuspidal representations: an exhaustion theorem
• Local geometric Langlands correspondence and affine Kac-Moody algebras

Related

• Local Jacquet-Langlands correspondence for regular supercuspidal representations
• Local Langlands correspondence for regular supercuspidal representations of GL(n)
• Geometric approach to the explicit local Langlands correspondence
• Simple supercuspidals and the Langlands correspondence
• Regular supercuspidal representations