# Stratified Langlands duality in the $A_{n}$ tower

### Graham A. Niblo

University of Southampton, UK### Roger Plymen

University of Southampton, UK### Nick Wright

University of Southampton, UK

## 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 $T_{n/k}//W$. 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 $K$-theory.

## Cite this article

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

DOI 10.4171/JNCG/315