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A new linear quotient of C 4 admitting a symplectic resolution

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We show that the quotient C 4/G admits a symplectic resolution for \({G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}\). Here Q 8 is the quaternionic group of order eight and D 8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation \({{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}\). This group is also naturally a subgroup of the wreath product group \({Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}\). We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C 4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Room 2.233, Alan Turing Building, Oxford Road, Manchester, M13 9PL, UK

    Gwyn Bellamy

  2. MIT Department of Mathematics, Room 2-172, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Travis Schedler

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  1. Gwyn Bellamy
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  2. Travis Schedler
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Correspondence to Travis Schedler.

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Bellamy, G., Schedler, T. A new linear quotient of C 4 admitting a symplectic resolution. Math. Z. 273, 753–769 (2013). https://doi.org/10.1007/s00209-012-1028-6

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  • DOI: https://doi.org/10.1007/s00209-012-1028-6

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