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Wilson’s Grassmannian and a noncommutative quadric. (English) Zbl 1059.58006
Introduction: Let the group $$\mu_m$$ of $$m$$th roots of unity act on the complex line by multiplication. This gives a $$\mu_m$$-action on Diff, the algebra of polynomial differential operators on the line. Following W. Crawley-Boevey and M. P. Holland [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)], we introduce a multiparameter deformation $$D_\tau$$ of the smash product $$\text{Diff}\#\mu_m$$. Our main result provides natural bijections between (roughly speaking) the following spaces:
(1) $$\mu_m$$-equivariant version of Wilson’s adelic Grassmannian of rank $$r$$;
(2) rank $$r$$ projective $$D_\tau$$-modules (with generic trivialization data);
(3) rank $$r$$ torsion-free sheaves on a “noncommutative quadric” $$\mathbb{P}^1\times_\tau\mathbb{P}^1$$;
(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 $$\mathcal D$$-modules and sheaves. The bijections between (2), (3), and (4) were motivated by our previous work [Compos. Math. 134, No. 3, 283-318 (2002; Zbl 1048.14001)]. The resulting bijection between (1) and (4) reduces, in the very special case: $$r=1$$ and $$\mu_m=\{1\}$$, 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 G. Wilson’s result [Invent. Math. 133, No. 1, 1-41 (1998; Zbl 0906.35089)].

##### MSC:
 58B25 Group structures and generalizations on infinite-dimensional manifolds 16S32 Rings of differential operators (associative algebraic aspects) 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14A22 Noncommutative algebraic geometry 14M15 Grassmannians, Schubert varieties, flag manifolds
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