Stratified Langlands duality in the A_n tower

Let S k denote a maximal torus in the complex Lie group G=SL n (C)/C k and let T k denote a maximal torus in its compact real form SU n (C)/C k , where k divides n . Let W denote the Weyl group of G , namely the symmetric group S n . We elucidate the structure of the extended quotient 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 nk //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.