Duke Mathematical Journal

The elliptic Hall algebra and the K-theory of the Hilbert scheme of A2

Olivier Schiffmann and 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 A2. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of GL. 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 P2, 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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1359036936

Digital Object Identifier
doi:10.1215/00127094-1961849

Mathematical Reviews number (MathSciNet)
MR3018956

Zentralblatt MATH identifier
1290.19001

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]

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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