Geometric categorifications of Verma modules: Grassmannian Quiver Hecke algebras

TL;DR

This paper develops a geometric framework for categorifying Verma modules using Grassmannian-Steinberg quiver flag varieties, extending previous algebraic and geometric approaches and providing new insights into their basis theorems.

Contribution

It introduces Grassmannian-Steinberg quiver flag varieties to geometrically realize Naisse-Vaz extension of KLR algebras for Verma modules, generalizing prior constructions.

Findings

Geometric realization of Naisse-Vaz algebras as convolution algebras in Borel-Moore homology

Explicit geometric dg-model of nil-Hecke algebras in the sl_2 case

New stratifications and diagram varieties explaining algebraic basis theorems

Abstract

Naisse and Vaz defined an extension of KLR algebras to categorify Verma modules. We realise these algebras geometrically as convolution algebras in Borel-Moore homology. For this we introduce Grassmannian-Steinberg quiver flag varieties. They generalize Steinberg quiver flag varieties in a non-obvious way, reflecting the diagrammatics from the Naisse-Vaz construction. Using different kind of stratifications we provide geometric explanations of the rather mysterious algebraic and diagrammatic basis theorems. A geometric categorification of Verma modules was recently found in the special case of $\mathfrak{sl}_2$ by Rouquier. Rouquier's construction uses coherent sheaves on certain quasi-map spaces to flag varieties (zastavas), whereas our construction is implicitly based on perverse sheaves. Both should be seen as parts (on dual sides) of a general geometric framework for the Naisse-Vaz approach. We first treat the (substantially easier) $\mathfrak{sl}_2$ case in detail and construct as a byproduct a geometric dg-model of the nil-Hecke algebras. The extra difficulties we encounter in general require the use of more complicated Grassmannian-Steinberg quiver flag varieties. Their definition arises from combinatorially defined diagram varieties which we assign to each Naisse-Vaz basis diagram. Our explicit analysis here might shed some light on categories of coherent sheaves on more general zastava spaces studied by Feigin-Finkelberg-Kuznetsov-Mirkovi\'c and Braverman, which we expect to occur in a generalization of Rouquier's construction away from $\mathfrak{sl}_2$.