Abstract:For a Dynkin quiver $Q$ (of type $\mathrm {ADE}$), 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 $\mathrm {E}$.

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中文翻译：

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Dynkin 箭袋类型的仿射最高权重类别和量子仿射 Schur-Weyl 对偶

摘要：对于Dynkin quiver $Q$（类型为$\mathrm {ADE}$），我们考虑某个Steinberg 类型分级箭袋变体的等变$K$-群的卷积代数的中心完成。我们观察到它是仿射拟遗传的，并证明它的有限维模块类别是在量子环代数 $U_ 上用 Hernandez-Leclerc 的幺半群模块 $\mathcal {C}_{Q}$ 的块标识的{q}(L\mathfrak {g})$ 通过 Nakajima 的同态。作为一个应用，我们证明了 Kang-Kashiwara-Kim 的广义量子仿射 Schur-Weyl 对偶函子在与 $Q$ 相关的 quiver Hecke 代数上的有限维模块的类别和 Hernandez-Leclerc 的类别 $\mathcal { C}_{Q}$，假设类型 $\mathrm {E}$ 的归一化 $R$-矩阵的一些极点的简单性。

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