Perverse sheaves on symmetric products of the plane

Abstract

For any field $k$, we give an algebraic description of the category ${\mathrm{Perv}}_{\mathcal{S}}(S^n({\mathbb{C}}^2),k)$ of perverse sheaves on the $n$-fold symmetric product of the plane $S^n({\mathbb{C}}^2)$ constructible with respect to its natural stratification and with coefficients in $k$. In particular, we show that it is equivalent to the category of modules over a new algebra that is closely related to the Schur algebra. As part of our description we obtain an analogue of modular Springer theory for the Hilbert scheme ${\mathrm{Hilb}}^n({\mathbb{C}}^2)$ of $n$ points in the plane with its Hilbert-Chow morphism.

50 pages. Added some clarifications and made minor corrections