Quiver maps, nilpotent orbits and Special Pieces of nilcones

A bstract This paper explores 3d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 quiver gauge theories whose moduli spaces represent nilpotent orbits, Słodowy slices or, more generally, Słodowy intersections, which span the Special Pieces of nilcones of Classical or Exceptional algebras. We introduce a map between magnetic and electric quivers containing symmetric group actions, such as wreathings (or loops), bouquets, and/or non-simply laced foldings, which can be related to symmetric subgroups of Lusztig’s canonical quotient groups for Special Pieces. This map on quivers induces a new map on nilpotent orbits that partially resolves the obstruction to quiver dualities presented by the non-involutive nature of the Lusztig Spaltenstein and Barbasch Vogan maps. We use Coulomb and Higgs branch quiver methods complemented by localisation formulae for verification. Some new quivers for intersections within Exceptional nilcones are presented.