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

Details 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 $\mathbb{A}^2$

2009-05-15

arxiv.org/abs/0905.2555v3

Mathematics

Quantum Algebra

57 pages, Latex, some minor corrections; a section added on Casimir operators and Heisenberg subalgebras; more details on the

Scientific paper

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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