Wilson’s Grassmannian and a Noncommutative Quadric

Abstract

Let the group μm of m th roots of unity act on the complex line by multiplication. This gives a μm-action on Diff, the algebra of polynomial differential operators on the line. Following Crawley-Boevey and Holland (1998), we introduce a multiparameter deformation Dτ of the smash product Diff #μm. Our main result provides natural bijections between (roughly speaking) the following spaces: (1) μm-equivariant version of Wilson's adelic Grassmannian of rank r ; (2) rank r projective Dτ-modules (with generic trivialization data); (3) rank r torsion-free sheaves on a “noncommutative quadric” ℙ1 × τℙ1; (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 Quiver varieties and a noncommutative ℙ2 (2002). The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and μm={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 (1998).