Poisson Traces for Symmetric Powers of Symplectic Varieties
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Etingof_Poisson traces.pdf
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396.9 KB
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Author(s)
Etingof, Pavel I.
Schedler, Travis
Date Issued
March 2013
Journal
International Mathematics Research Notices
Publisher
Oxford University Press
Citation
Etingof, P., and T. Schedler. “Poisson Traces for Symmetric Powers of Symplectic Varieties.” International Mathematics Research Notices (March 21, 2013).
Version
Author's final manuscript
Abstract
We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities T*C[superscript n-1]/S[subscript n] associated with the type A Weyl group S[subscript n] and its reflection representation C[superscript n-1]. We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional irreducible representations and prime ideals of quantizations of these varieties. Finally, motivated by these results, we pose conjectures on symplectic resolutions, and give related examples of the natural D-module. In an appendix, the second author computes the Poisson traces and associated D-module for the quotients T*C[superscript n]/D[subscript n] associated with type D Weyl groups. In a second appendix, the same author provides a direct proof of one of the main theorems.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/imrn/rnt031
