Annals of K-Theory

A fixed point theorem on noncompact manifolds

Peter Hochs and Hang Wang

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We generalise the Atiyah–Segal–Singer fixed point theorem to noncompact manifolds. Using K K -theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah–Segal–Singer’s result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant K -theory and K -homology, and a nonlocalised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.

Article information

Ann. K-Theory, Volume 3, Number 2 (2018), 235-286.

Received: 14 October 2016
Revised: 19 September 2017
Accepted: 4 October 2017
First available in Project Euclid: 4 April 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]
Secondary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 22E46: Semisimple Lie groups and their representations

equivariant index fixed point formula noncompact manifold $KK$-theory


Hochs, Peter; Wang, Hang. A fixed point theorem on noncompact manifolds. Ann. K-Theory 3 (2018), no. 2, 235--286. doi:10.2140/akt.2018.3.235.

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