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Stratified Langlands duality in the An tower

Stratified Langlands duality in the An tower
Stratified Langlands duality in the An tower
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.
K-theory, representation theory, Langlands duality
1661-6952
193-225
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944
Plymen, Roger J
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Wright, Nicholas
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944
Plymen, Roger J
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Wright, Nicholas
f4685b8d-7496-47dc-95f0-aba3f70fbccd

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

Record type: Article

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.

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Submitted date: 13 February 2017
Accepted/In Press date: 27 January 2018
e-pub ahead of print date: 5 January 2019
Keywords: K-theory, representation theory, Langlands duality

Identifiers

Local EPrints ID: 418303
URI: https://eprints.soton.ac.uk/id/eprint/418303
ISSN: 1661-6952
PURE UUID: 7b559aa2-5e41-4cd0-aee9-7ef72984806e
ORCID for Graham Niblo: ORCID iD orcid.org/0000-0003-0648-7027
ORCID for Nicholas Wright: ORCID iD orcid.org/0000-0003-4884-2576

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Date deposited: 27 Feb 2018 17:31
Last modified: 20 Jul 2019 04:29

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Contributors

Author: Graham Niblo ORCID iD
Author: Roger J Plymen
Author: Nicholas Wright ORCID iD

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