Duke Mathematical Journal

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

Abstract

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^{2}$. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of $\mathrm{GL}_{\infty}$. 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 $\mathbb{P}^{2}$, virtual fundamental classes, shuffle algebras, …).

Article information

Source
Duke Math. J., Volume 162, Number 2 (2013), 279-366.

Dates
First available in Project Euclid: 24 January 2013

https://projecteuclid.org/euclid.dmj/1359036936

Digital Object Identifier
doi:10.1215/00127094-1961849

Mathematical Reviews number (MathSciNet)
MR3018956

Zentralblatt MATH identifier
1290.19001

Citation

Schiffmann, Olivier; Vasserot, Eric. The elliptic Hall algebra and the $K$ -theory of the Hilbert scheme of $\mathbb{A}^{2}$. Duke Math. J. 162 (2013), no. 2, 279--366. doi:10.1215/00127094-1961849. https://projecteuclid.org/euclid.dmj/1359036936

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