[Paper Review] Equivariant Coherent Sheaves, Soergel Bimodules, and Categorification of Affine Hecke Algebras

Christopher Stephen Dodd|arXiv (Cornell University)|Aug 19, 2011

Algebraic structures and combinatorial models20 references12 citations

TL;DR

This paper establishes a categorical equivalence between equivariant coherent sheaves on Grothendieck's resolution and the categorical affine Hecke algebra via Soergel bimodules, generalizing Kazhdan-Lusztig's geometric construction. It proves that the weak braid group action on these sheaf categories can be upgraded to a strict action, providing a categorification of affine Hecke algebras with enhanced algebraic structure.

ABSTRACT

We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf categories to the categories of perverse sheaves considered in the work of Bezrukavnikov-Yun, generalizing results of Arkhipov-Bezrukavnikov. In addition, we deduce that the weak braid group action on sheaves of Riche and Bezrukavnikov-Riche can be upgraded to a strict braid group action.

Motivation & Objective

• To establish a geometric categorification of the affine Hecke algebra using equivariant coherent sheaves on Grothendieck's resolution.
• To relate the category of equivariant coherent sheaves to the categorical affine Hecke algebra via Soergel bimodules.
• To upgrade the weak braid group action on coherent sheaves to a strict braid group action.
• To generalize Kazhdan-Lusztig's equivalence to the affine setting and connect it with perverse sheaf categories.
• To demonstrate that the perversely exotic t-structure on coherent sheaves corresponds to the heart of the perverse t-structure in the Dwork-Weil setting.

Proposed method

• Construct a functor κ from the derived category of G×Gm-equivariant coherent sheaves on the enhanced nilpotent cone to the derived category of graded modules over the categorical affine Hecke algebra.
• Use Kostant-Whittaker reduction to relate the geometry of the enhanced nilpotent cone to the structure of Soergel bimodules.
• Apply deformation theory to the flag variety and its resolution, introducing a formal parameter h to define the category of D_h-modules and coherent sheaves on the enhanced resolution.
• Utilize Fourier-Mukai transforms with kernels defined via deformed reflection functors to construct braid group actions on the derived categories.
• Leverage the existence of tilting generators and t-structures (especially the perversely exotic t-structure) to control the category and prove equivalence.
• Show that the braid group action lifts from weak to strict by constructing preferred isomorphisms between convolutions of different braid word decompositions via the functor κ and its compatibility with convolution.

Experimental results

Research questions

• RQ1 How can the affine Hecke algebra be categorically realized via equivariant coherent sheaves on Grothendieck's resolution?
• RQ2 What is the precise relationship between the category of equivariant coherent sheaves and the categorical affine Hecke algebra of Soergel?
• RQ3 Can the weak braid group action on coherent sheaves be upgraded to a strict braid group action?
• RQ4 How does the perversely exotic t-structure on coherent sheaves relate to the perverse t-structure on Dwork-Weil sheaves?
• RQ5 To what extent does the Kostant-Whittaker reduction functor κ preserve the categorical structure and induce an equivalence?

Key findings

• The functor κ induces an equivalence between the derived category of G×Gm-equivariant coherent sheaves on the enhanced nilpotent cone and the derived category of graded modules over the categorical affine Hecke algebra.
• The weak braid group action on the category of equivariant coherent sheaves lifts to a strict braid group action, with isomorphisms between different convolution decompositions being canonical and functorial.
• The perversely exotic t-structure on the category of coherent sheaves corresponds to the heart of the perverse t-structure in the Dwork-Weil setting, confirming a conjectural equivalence.
• The Kostant-Whittaker reduction functor κ preserves the braid group action and induces a strict action on the target category of Soergel bimodules.
• The equivalence constructed here matches the one in [AB] on objects, and the braid group action on perverse sheaves via convolution with tilting sheaves is isomorphic to the action lifted from coherent sheaves.
• The derived category of D_h-modules on the enhanced flag variety admits a strict braid group action, extending to ungraded categories via restriction.