Geometric categorifications of Verma modules: Grassmannian Quiver Hecke algebras
Résumé
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 sl2 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) sl2 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-Mirković and Braverman, which we expect to occur in a generalization of Rouquier's construction away from sl2.
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