Spherical Hall algebras of a weighted projective curve

Abstract: In this article, we deal with the structure of the spherical Hall algebra of coherent sheaves with parabolic structures on a smooth projective curve of arbitrary genus. We provide a shuffle-like presentation of the vector bundle part and show the existence of the generic form. We also prove that the spherical Hall algebra contains the characteristic functions on all the Harder-Narasimhan strata. These results together imply Schiffmann's theorem on the existence of Kac polynomials for quasi-parabolic vector bundles of fixed rank and multi-degree over the curve. On the other hand, the shuffle structure we obtained is new and we make links to the representations of quantum affine algebras of type A.