The Hall algebra of the category of coherent sheaves on an elliptic curve defined over a finite field has been explicitly described and shown to be a 2-parameter deformation of the ring of diagonal invariants (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 . This allows us to define a canonical basis of 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.
"On the Hall algebra of an elliptic curve, II." Duke Math. J. 161 (9) 1711 - 1750, 15 June 2012. https://doi.org/10.1215/00127094-1593362