2024/05/30 by Ruslan Maksimau, Maksimau, Ruslan, Catharina Stroppel +1
Mathematics · #14M15 #16S30 #17B10 #17B20 #20G05 #55N25 #55N33 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2405.20262
openalex publication_date 2024/05/30 · openalex created_date 2024/05/31 · openalex updated_date 2026/07/28
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 \mathfraksl2 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) \mathfraksl2 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 \mathfraksl2.