Niblo, Graham, Plymen, Roger J and Wright, Nicholas
(2019)
Stratified Langlands duality in the A_{n} tower.
*Journal of Noncommutative Geometry*, 13 (1), 193-225.
(doi:10.4171/JNCG/315).

## Abstract

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_n //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.

**stratifiedlanglandsRevisedPlymenEdits - Accepted Manuscript**

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- Current Faculties > Faculty of Social Sciences > School of Mathematical Sciences > Pure Mathematics

School of Mathematical Sciences > Pure Mathematics - Faculties (pre 2018 reorg) > Faculty of Social, Human and Mathematical Sciences (pre 2018 reorg) > Mathematical Sciences (pre 2018 reorg) > Pure Mathematics (pre 2018 reorg)

Current Faculties > Faculty of Social Sciences > School of Mathematical Sciences > Mathematical Sciences (pre 2018 reorg) > Pure Mathematics (pre 2018 reorg)

School of Mathematical Sciences > Mathematical Sciences (pre 2018 reorg) > Pure Mathematics (pre 2018 reorg)

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