Stratified Langlands duality in the $A_n$ tower

Let ${\mathbf{S}}_k$ denote a maximal torus in the complex Lie group $\mathbf{G}={\mathrm{S}\mathrm{L}}_n(\mathbb{C})\mathrm{/}{C}_k$ and let $T_k$ denote a maximal torus in its compact real form ${\mathrm{S}\mathrm{U}}_n(\mathbb{C})\mathrm{/}{C}_k$, where $k$ divides $n$. Let $W$ denote the Weyl group of $\mathbf{G}$, namely the symmetric group ${\mathfrak{S}}_n$. We elucidate the structure of the extended quotient ${\mathbf{S}}_k\mathrm{/}\mathrm{/}W$ as an algebraic variety and of $T_k\mathrm{/}\mathrm{/}W$ as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of $K$-theory under Langlands duality, this calculation provides a homotopy equivalence between $T_k\mathrm{/}\mathrm{/}W$ and its dual $T_{\frac{n}{k}}\mathrm{/}\mathrm{/}W$. Hence there is an isomorphism in cohomology for the extended quotients which is stratified as a direct sum over conjugacy classes of the Weyl group. We use our formula to compute a number of examples.