Wilson's grassmannian and a noncommutative Quadric
Loading...
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
Let the group G of m-th roots of unity act on the complex line by multiplication, inducing an action on the algebra, Diff, of polynomial differential operators on the line. Following Crawley-Boevey and Holland, we introduce a multiparameter deformation, D_c, of the smash-product (Diff # G). Our main result provides natural bijections between (roughly speaking) the following spaces:
(1) G-equivariant version of Wilson's adelic Grassmannian of rank r;
(2) Rank r projective D_c-modules (equipped with generic trivialization);
(3) Rank r torsion-free sheaves on a `noncommutative quadric';
(4) Disjoint union of Nakajima quiver varieties for the cyclic quiver with m vertices. The bijection between (1) and (2) is provided by a version of Riemann-Hilbert correspondence between D-modules and sheaves. The bijections between (2), (3) and (4) were motivated by our previous work math.AG/0103068. The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and G=1, to the partition of (rank 1) adelic Grassmannian into a union of Calogero-Moser spaces, discovered by Wilson. This gives, in particular, a natural and purely algebraic approach to Wilson's result.
35pp, several remarks, and an open problem (at the end of the Introduction) added
35pp, several remarks, and an open problem (at the end of the Introduction) added