Vol. 3, No. 2, 2018

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A fixed point theorem on noncompact manifolds

Peter Hochs and Hang Wang

Vol. 3 (2018), No. 2, 235–286

We generalise the Atiyah–Segal–Singer fixed point theorem to noncompact manifolds. Using KK-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.

equivariant index, fixed point formula, noncompact manifold, $KK$-theory
Mathematical Subject Classification 2010
Primary: 58J20
Secondary: 19K35, 22E46
Received: 14 October 2016
Revised: 19 September 2017
Accepted: 4 October 2017
Published: 24 March 2018
Peter Hochs
School of Mathematical Sciences
University of Adelaide
North Terrace Campus
Adelaide, SA
Hang Wang
Research Center for Operator Algebras
East China Normal University
School of Mathematical Sciences
University of Adelaide
North Terrace Campus
Adelaide, SA