On the Hall algebra of an elliptic curve, II

Abstract

The Hall algebra $U_E^{+}$ of the category of coherent sheaves on an elliptic curve $E$ defined over a finite field has been explicitly described and shown to be a 2-parameter deformation of the ring of diagonal invariants $R^{+}=C[x_1^{\pm 1},\dots ,y_1,\dots {]}^{S_{\infty }}$ (in infinitely many variables). We study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on $E$. This allows us to define a canonical basis $B$ of $U_E^{+}$ in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice, and a certain PBW-type basis.