  One can associate an invariant to a large class of regular codimension two
defects of the six dimensional $(0,2)$ SCFT $\mathscr{X}[j]$ using the
classical Springer correspondence. Such an association allows a simple
description of S-duality of associated Gaiotto-Witten boundary conditions in
$\mathcal{N}=4$ SYM for arbitrary gauge group and by extension, a determination
of certain local aspects of class $\mathcal{S}$ constructions. I point out that
the problem of \textit{classifying} the corresponding boundary conditions in
$\mathcal{N}=4$ SYM is intimately tied to possible symmetry breaking patterns
in the bulk theory. Using the Springer correspondence and the representation
theory of Weyl groups, I construct a pair of functors between the class of
boundary conditions in the theory in the phase with broken gauge symmetry and
those in the phase with unbroken gauge symmetry.
