The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$
The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$
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DOI：
10.1215/00127094-1961849
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发表时间：
2009-05
期刊：
影响因子：
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通讯作者：
O. Schiffmann;E. Vasserot
中科院分区：
文献类型：
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作者：
O. Schiffmann;E. Vasserot
在本文中，我们计算A^2的希尔伯特格式的等变K-理论中的卷积代数。我们证明了它同构于椭圆Hall代数，从而同构于GL_infty的球面DAHA.我们通过椭圆曲线的几何朗兰兹对应来解释这种巧合，通过将其与交换簇的等变K-理论中的卷积代数联系起来。我们还得到了其他一些相关的结果（椭圆霍尔代数对P^2上框架无挠层的模空间的K-理论的作用，虚基本类，shuffle代数，.）。
In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2. We show that it is isomorphic to the elliptic Hall algebra, and hence to the spherical DAHA of GL_\infty. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over P^2, virtual fundamental classes, shuffle algebras,...).