Perverse coherent sheaves on symplectic singularities
Ilya Dumanski
TL;DR
This work introduces perverse coherent (Poisson) sheaves on conical symplectic singularities by employing Harish-Chandra Lie algebroids and their modules, constructing a perverse t-structure under suitable orbit-codimension assumptions. It defines a category of perverse coherent sheaves on symplectic singularities via the Poisson cotangent algebroid Ω_X and analyzes simple objects as IC-extensions from symplectic leaves, with explicit classification in the HC setting. The authors prove that simple objects form a basis in the Grothendieck group and verify this in key cases, notably the nilpotent cone and affine Grassmannian slices, where the bases recover known canonical bases and are compatible with K-theory. The paper also outlines a broad program for lifting these structures to symplectic resolutions, relating to Lusztig/Kazhdan–Lusztig bases and stable envelopes, and suggests extensions to Slodowy slices and double affine Grassmannians. Overall, the results provide a unified Lie-algebroid–Poisson framework for canonical bases in a wide class of symplectic singularities with potential links to quantum and geometric representation theory.
Abstract
We propose the notion of perverse coherent sheaves for symplectic singularities and study its properties. In particular, it gives a basis of simple objects in the Grothendieck group of Poisson sheaves. We show that perverse coherent bases for the nilpotent cone and for the affine Grassmannian arise as particular cases of our construction.