Journal of Noncommutative Geometry

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Volume 11, Issue 3, 2017, pp. 1001–1036
DOI: 10.4171/JNCG/11-3-7

Published online: 2017-09-26

Langlands functorality in $K$-theory for $C^*$-algebras. I. Base change

Kuok Fai Chao[1] and Hang Wang[2]

(1) Shanghai University, China
(2) University of Adelaide, Australia and East China Normal University, Shanghai, China

We compare representations of the real and complex general linear groups and special linear groups in the framework of $K$-theory, using base change on $L$-parameters. We introduce a notion of base change on $K$-theory involving the fixed point set of the reduced dual of a complex group. For general linear groups, we prove that the base change map is compatible with the Connes–Kasparov isomorphism.

Keywords: $K$-theory, local Langlands correspondence, base change, reduced group $C^*$-algebra, tempered representation

Chao Kuok Fai, Wang Hang: Langlands functorality in $K$-theory for $C^*$-algebras. I. Base change. J. Noncommut. Geom. 11 (2017), 1001-1036. doi: 10.4171/JNCG/11-3-7