Stratified Langlands duality in the $A_n$ tower

Abstract

Let ${\mathbf{S}}_k$ denote a maximal torus in the complex Lie group $\mathbf{G}=S{L}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal torus in its compact real form $S{U}_n(\mathbb{C})/C_k$, 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 ${\mathbf{S}}_k//W$ as an algebraic variety and of $T_k//W$ 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 $T_k//W$ and its dual $T_{n/k}//W$. Hence there is an isomorphism in cohomology for the extended quotients. Moreover this is stratified as a direct sum over conjugacy classes of the Weyl group. We derive a formula for the periodic cyclic homology of the group ring of an extended affine Weyl group in terms of these extended quotients and use our formulae to compute a number of examples of homology, cohomology and $K$-theory.