Equivariant K-theory, nested Hilbert schemes and virtual classes

Abstract : We consider a convolution algebra $\mathbf{E}$ in the
equivariant K-theory of the Hilbert schemes of points in the plane. We
show that the subalgebra $\mathbf{H} \subset \mathbf{E}$ generated by the
powers of the tautological bundles on smooth nested Hilbert schemes $Z_{n,
n \pm 1}$ is isomorphic to the so-called Hall algebra of an elliptic
curve. The natural basis of that Hall algebra (conjecturally) corresponds
to certain tautological bundles on the \textit{virtual} classes of the
nested Hilbert schemes $Z_{n,n \pm k}$ (for \textit{any} $k$). These
virtual classes were recently computed by Carlsson and Okounkov. We also
give a possible interpretation in terms of the geometric Langlands
program. This is joint work with E. Vasserot.