JournalsjncgVol. 13, No. 1pp. 193–225

Stratified Langlands duality in the AnA_n 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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Abstract

Let Sk\mathbf S_k denote a maximal torus in the complex Lie group G=SLn(C)/Ck\mathbf G = SL_n(\mathbb C)/C_k and let TkT_k denote a maximal torus in its compact real form SUn(C)/CkSU_n(\mathbb C)/C_k, where kk divides nn. Let WW denote the Weyl group of G\mathbf G, namely the symmetric group Sn\mathfrak S_n. We elucidate the structure of the extended quotient Sk/ ⁣/W\mathbf S_k {/\!/} W as an algebraic variety and of Tk/ ⁣/WT_k{/\!/} W as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of KK-theory under Langlands duality, this calculation provides a homotopy equivalence between Tk/ ⁣/WT_k{/\!/} W and its dual Tn/k/ ⁣/WT_{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 KK-theory.

Cite this article

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

DOI 10.4171/JNCG/315