Duke Mathematical Journal

On the Hall algebra of an elliptic curve, II

Olivier Schiffmann

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The Hall algebra UE+ 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[x1±1,,y1,]S (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 UE+ 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.

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Duke Math. J., Volume 161, Number 9 (2012), 1711-1750.

First available in Project Euclid: 6 June 2012

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Zentralblatt MATH identifier

Primary: 22E57: Geometric Langlands program: representation-theoretic aspects [See also 14D24]
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]


Schiffmann, Olivier. On the Hall algebra of an elliptic curve, II. Duke Math. J. 161 (2012), no. 9, 1711--1750. doi:10.1215/00127094-1593362. https://projecteuclid.org/euclid.dmj/1338987167

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