Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

Abstract

For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category ${\mathcal{C}}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with $Q$ and Hernandez-Leclerc's category ${\mathcal{C}}_Q$, assuming the simpleness of some poles of normalized R-matrices for type E.

v2: 51 pages, a minor revision. The proof of Lemma 2.24 has been corrected. v3: 52 pages, the previous Subsection 5.4 is deleted, expositions are improved

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• 2-Kac-Moody algebras
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• Quiver Varieties and Branching
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Cited by in corpus (8)

• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Geometric realization of Dynkin quiver type quantum affine Schur-Weyl duality
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras
• Equivariant multiplicities via representations of quantum affine algebras
• Braid group action on the module category of quantum affine algebras
• Strands algebras and the affine highest weight property for equivariant hypertoric categories