In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of . We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of . 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 , virtual fundamental classes, shuffle algebras, …).
Olivier Schiffmann. Eric Vasserot. "The elliptic Hall algebra and the -theory of the Hilbert scheme of ." Duke Math. J. 162 (2) 279 - 366, 1 February 2013. https://doi.org/10.1215/00127094-1961849