Hot Topics: Geometric Langlands Conjecture: The stack of local systems and ind-coherent sheaves
Presenter
April 6, 2026
Keywords:
- geometric Langlands
- Opers
- Kac–Moody algebras
- Factorization algebras
- Factorization homology
- affine Grassmannian
- Automorphic sheaves
- Whittaker
- Eisenstein series
- moduli spaces
MSC:
- 14D24 - Geometric Langlands program (algebro-geometric aspects)
- 22E57 - Geometric Langlands program: representation-theoretic aspects
- 11G45 - Geometric class field theory
- 14F08 - Derived categories of sheaves dg categories and related constructions in algebraic geometry
- 14F10 - Differentials and other special sheaves D-modules Bernstein-Sato ideals and polynomials
- 14H60 - Vector bundles on curves and their moduli
- 14M15 - Grassmannians Schubert varieties flag manifolds
- 17B10 - Representations of Lie algebras and Lie superalgebras algebraic theory (weights)
- 17B67 - Kac-Moody (super)algebras extended affine Lie algebras toroidal Lie algebras
- 17B69 - Vertex operators vertex operator algebras and related structures
- 22E47 - Representations of Lie and real algebraic groups algebraic methods (Verma modules
- etc.)
- 22E67 - Loop groups and related constructions group-theoretic treatment
- 81R10 - Infinite-dimensional groups and algebras motivated by physics including Virasoro Kac-Moody $W$W-algebras and other current algebras and their representations
Abstract
The goal of this talk is to formalize the spectral side of the geometric Langlands equivalence. We define the stack LocSys of de Rham local systems and prove its basic properties. Beilinson and Drinfeld's "best hope" for the geometric Langlands equivalence uses quasi-coherent sheaves on LocSys, but that category is too small for the equivalence. We therefore enlarge QCoh(LocSys) to the category of ind-coherent sheaves on LocSys with nilpotent singular support.