A new linear quotient of C^4 admitting a symplectic resolution

TL;DR

This paper demonstrates that the quotient of C^4 by a specific group G admits a symplectic resolution, providing explicit computations and exploring broader classification questions for such quotients.

Contribution

It introduces a new example of a linear quotient admitting a symplectic resolution and analyzes its singularities and deformation theory.

Findings

C^4/G admits a symplectic resolution.

Computed the singular locus of related symplectic reflection algebras.

Discussed broader classification of linear quotients with symplectic resolutions.

Abstract

We show that the quotient C^4/G admits a symplectic resolution for G = (Q_8 x D_8)/(Z/2) < Sp(4,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 -1 of each. It is equipped with the tensor product of the defining two-dimensional representations of Q_8 and D_8. This group is also naturally a subgroup of the wreath product group of Q_8 by S_2. 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.