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Wilson’s Grassmannian and a Noncommutative Quadric

Baranovsky, Vladimir and Ginzburg, Victor and Kuznetsov, Alexander (2003) Wilson’s Grassmannian and a Noncommutative Quadric. International Mathematics Research Notices, 2003 (21). pp. 1155-1197. ISSN 1073-7928.

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Let the group μ_m of m th roots of unity act on the complex line by multiplication. This gives a μ_m-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 #μ_m. Our main result provides natural bijections between (roughly speaking) the following spaces: (1) μ_m-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” ℙ^1 × τℙ^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 D-modules and sheaves. The bijections between (2), (3), and (4) were motivated by our previous work Quiver varieties and a noncommutative ℙ^2 (2002). The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and μ_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 Wilson's result (1998).

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Additional Information:© 2003 Hindawi Publishing Corporation. Received 17 October 2002. Accepted March 2, 2003. To Yuri Ivanovich Manin on the occasion of his 65th birthday. Communicated by Yuri I. Manin. We are indebted to Sasha Beilinson for some very useful remarks. The third author was partially supported by RFFI grants 99-01-01144 and 99-01-01204 INTAS-PEN-2000-269. This work was made possible in part by CRDFA ward No. RM1-2406-MO-02.Also, he would like to express his gratitude to the University of Chicago, where the major part of this paper was written.
Funding AgencyGrant Number
Russian Foundation for Fundamental Investigations (RFFI) 99-01-01144
Russian Foundation for Fundamental Investigations (RFFI) 99-01-01204
CRDF AwardRM1-2406-MO-02
Record Number:CaltechAUTHORS:20111021-080537056
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Official Citation: Vladimir Baranovsky, Victor Ginzburg, and Alexander Kuznetsov Wilson's Grassmannian and a noncommutative quadric Int Math Res Notices (2003) Vol. 2003 1155-1197 doi:10.1155/S1073792803210126
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27342
Deposited By: Ruth Sustaita
Deposited On:21 Oct 2011 15:41
Last Modified:23 Aug 2016 10:06

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