Stratified Langlands duality in the Antower Niblo, Graham and Plymen, Roger and Wright, Nick 2016

Let Sk denote a maximal torus in the complex Lie group G = SLn(C)/Ck and let Tk denote a maximal torus in its compact real form SUn(C)/Ck, where k divides n. Let W denote the Weyl group of G, namely the symmetric group Sn. We elucidate the structure of the extended quotient Sk//W as an algebraic variety and of Tk//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 Tk//W and its dual Tn k //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.