Stratified Langlands duality in the tower

  • Graham A. Niblo

    University of Southampton, UK
  • Roger Plymen

    University of Southampton, UK
  • Nick Wright

    University of Southampton, UK
Stratified Langlands duality in the $A_n$ tower cover

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Let denote a maximal torus in the complex Lie group and let denote a maximal torus in its compact real form , where divides . Let denote the Weyl group of , namely the symmetric group . We elucidate the structure of the extended quotient as an algebraic variety and of as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of -theory under Langlands duality, this calculation provides a homotopy equivalence between and its dual . 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 -theory.

Cite this article

Graham A. Niblo, Roger Plymen, Nick Wright, Stratified Langlands duality in the tower. J. Noncommut. Geom. 13 (2019), no. 1, pp. 193–225

DOI 10.4171/JNCG/315