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K-theory, fixed point theorem and representation of semisimple Lie groups

Banff International Research Station for Mathematical Innovation and Discovery

K-theory of reduced group $C^*$-algebras and their trace maps can be used to study tempered representations of a semisimple Lie group from the point of view of index theory. For a semisimple Lie group, every K-theory generator can be viewed as the equivariant index of some Dirac operator, but also interpreted as a (family of) representation(s) parametrised by A in the Levi component of a cuspidal parabolic subgroup. In particular, if the group has discrete series representations, the corresponding K-theory classes can be realised as equivariant geometric quantisations of the associated coadjoint orbits. Applying orbital traces to the K-theory group, we obtain a fixed point formula which, when applied to this realisation of discrete series, recovers Harish-Chandra's character formula for the discrete series on the representation theory side. This is a noncompact analogue of Atiyah-Segal-Singer fixed point theorem in relation to the Weyl character formula. This is joint work with Peter Hochs.

Mathematics; Differential geometry; Topological groups, lie groups

video/mp4

Author affiliation: East China Normal University/University of Adelaide

2018-10-16

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67574

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/