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The local spectrum of the Dirac operator for the universal cover of SL_2(R)

The local spectrum of the Dirac operator for the universal cover of SL_2(R)
The local spectrum of the Dirac operator for the universal cover of SL_2(R)
Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of SL2(R)SL2(R), exhibiting it as the generator of KK1(C,A)KK1(C,A), where AA is the reduced C?C?-algebra of the group. This yields a new and direct computation of the K -theory of AA. A fundamental role is played by the limit-of-discrete-series representation, which is the frontier between the discrete and the principal series of the group. We provide a detailed analysis of the localised spectra of the Dirac operator and compute the Dirac cohomology.
universal cover of SL2(R), dirac operator, k-theory, dirac cohomology
0022-1236
957-975
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Plymen, Roger
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Plymen, Roger
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd

Brodzki, Jacek, Niblo, Graham A., Plymen, Roger and Wright, Nick (2015) The local spectrum of the Dirac operator for the universal cover of SL_2(R). Journal of Functional Analysis, 270 (3), 957-975. (doi:10.1016/j.jfa.2015.10.010).

Record type: Article

Abstract

Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of SL2(R)SL2(R), exhibiting it as the generator of KK1(C,A)KK1(C,A), where AA is the reduced C?C?-algebra of the group. This yields a new and direct computation of the K -theory of AA. A fundamental role is played by the limit-of-discrete-series representation, which is the frontier between the discrete and the principal series of the group. We provide a detailed analysis of the localised spectra of the Dirac operator and compute the Dirac cohomology.

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More information

Accepted/In Press date: 21 October 2015
e-pub ahead of print date: 28 November 2015
Published date: 28 November 2015
Keywords: universal cover of SL2(R), dirac operator, k-theory, dirac cohomology
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 369701
URI: https://eprints.soton.ac.uk/id/eprint/369701
ISSN: 0022-1236
PURE UUID: e96d0a57-d56f-4efb-8c5f-9c24e46c8bd3

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Date deposited: 13 Oct 2014 10:15
Last modified: 17 Jul 2017 21:55

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