A new linear quotient of C 4 admitting a symplectic resolution

Abstract

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.