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arXiv · 0905.2555

The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$

Olivier Schiffmann·Eric Vasserot
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Abstract

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,...).

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Olivier Schiffmann, Eric Vasserot. 2009-05-15. The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$. https://doi.org/10.1215/00127094-1961849

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