论文标题
Hikita-Nakajima猜想的Gieseker品种
Hikita-Nakajima conjecture for the Gieseker variety
论文作者
论文摘要
令$ \ mathfrak {M} _0 $为仿射中nakajima Quiver品种，而$ \ Mathcal {M} $是相应的BFN COULOMB分支。假设$ \ mathfrak {m} _0 $可以通过（平滑）nakajima颤抖品种$ \ mathfrak {m} $解决。 hikita-nakajima猜想声称应该存在（分级）代数$ h^*_ {s}（\ mathfrak {m}，\ mathbb {c}）\ simeq \ Mathbb {C} [\ Mathcal {M} _ {\ Mathfrak {s}}}}^{\ Mathbb {c}^\ times}] $，这里$ \ curvearrowrow \ mathfrak \ Mathfrak {M} $ \ MATHCAL {M} _ {\ MATHFRAK {s}} $是$ \ Mathcal {M} $ a $ \ Mathfrak {s} = \ operatatorName {lie}（lie}（s）$，$ \ mathbb {s）$，$ \ mathbb {c} c} portion的（poisson）的变形。 $ \ MATHCAL {M} $，和$ \ MATHBB {C} [\ MATHCAL {M} _ {\ MATHFRAK {\ MATHFRAK {s}}}^{\ MATHBB {C}^\ times}] $ $ \ MATHCAL {M} _ {\ MATHFRAK {S}} $。我们证明了$ \ mathfrak {m} = \ mathfrak {m} {m}（n，r）$ gieseker variew（$ adhm $ space）的hikita-nakajima猜想。我们明确地在发电机上产生同构。我们还使用$ \ Mathcal {M} _ {\ Mathfrak {s s}} $的实现来描述上面的Hikita-Nakajima同构，作为合理Cherednik Algebra中心的光谱，对应于$ s_n \ ltimes（Z Z}）在同构中出现的代数，其中心是归类性的循环hecke代数（概括了Shan，Varagnolo和Vasserot的一些结果）。
Let $\mathfrak{M}_0$ be an affine Nakajima quiver variety, and $\mathcal{M}$ is the corresponding BFN Coulomb branch. Assume that $\mathfrak{M}_0$ can be resolved by the (smooth) Nakajima quiver variety $\mathfrak{M}$. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras $H^*_{S}(\mathfrak{M},\mathbb{C}) \simeq \mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$, here $S \curvearrowright \mathfrak{M}_0$ is a torus acting on $\mathfrak{M}_0$ preserving the Poisson structure, $\mathcal{M}_{\mathfrak{s}}$ is the (Poisson) deformation of $\mathcal{M}$ over $\mathfrak{s}=\operatorname{Lie} (S)$, $\mathbb{C}^\times$ is a generic one-dimensional torus acting on $\mathcal{M}$, and $\mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$ is the algebra of schematic $\mathbb{C}^\times$-fixed points of $\mathcal{M}_{\mathfrak{s}}$. We prove the Hikita-Nakajima conjecture for $\mathfrak{M}=\mathfrak{M}(n,r)$ Gieseker variety ($ADHM$ space). We produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of $\mathcal{M}_{\mathfrak{s}}$ as the spectrum of the center of rational Cherednik algebra corresponding to $S_n \ltimes (\mathbb{Z}/r\mathbb{Z})^n$ and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).
