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

@article{Schiffmann2009TheEH, title={The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of \$\mathbb\{A\}^2\$}, author={Olivier Schiffmann and Eric Vasserot}, journal={arXiv: Quantum Algebra}, year={2009} }

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…

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