Noncommutative Counterparts of the Springer Resolution

Abstract

Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebro-geometric problems, such as the derived equivalence conjecture and description of T. Bridgeland's space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a $t$-structure on the derived category of the resolution. The intriguing fact that the same $t$-structure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.

ICM talk; 23 pages

References in corpus (9)

• Perverse sheaves on a Loop group and Langlands' duality
• Derived categories of coherent sheaves
• Localization of modules for a semisimple Lie algebra in prime characteristic
• Stability conditions on a non-compact Calabi-Yau threefold
• McKay equivalence for symplectic resolutions of singularities
• Nonabelian Hodge Theory in Characteristic p
• Fedosov quantization in positive characteristic
• Singular localization and intertwining functors for reductive Lie algebras in prime characteristic
• Derived equivalences by quantization

Cited by in corpus (24)

• The Character Theory of a Complex Group
• Koszul duality and modular representations of semi-simple Lie algebras
• Non-commutative desingularization of determinantal varieties, I
• Loop group actions on categories and Whittaker invariants
• Elliptic Springer Theory
• Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution
• Coherent Springer theory and the categorical Deligne-Langlands correspondence
• Noncommutative desingularization of orbit closures for some representations of $G{L}_n$
• Loop Spaces and Langlands Parameters
• Beilinson-Bernstein localization over the Harish-Chandra center
• Tilting bundles via the Frobenius morphism
• Affine tangles and irreducible exotic sheaves
• Orthogonally spherical objects and spherical fibrations
• Morita equivalence for convolution categories: Appendix to arXiv:0805.0157
• Stability conditions for Slodowy slices and real variations of stability
• Singular localization and intertwining functors for reductive Lie algebras in prime characteristic
• A Survey on Springer Theory
• On Singular Localization of $\mathfrak{g}$-modules
• On the structure of the affine asymptotic Hecke algebras
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• D-modules on rigid analytic spaces I
• Injectivity of a certain cycle map for finite dimensional W-algebras
• Exceptional collections, t-structures, and categorical action of shifted $q=0$ affine algebra, ${\mathfrak{sl}}_2$ case
• Groups actions, and D equivalences of categories of coherent sheaves of symplectic resolutions