Wilson’s Grassmannian and a noncommutative quadric.

*(English)*Zbl 1059.58006Introduction: 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)].

(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 |