2015/12/31 by Peter Hochs, Hang Wang
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Atiyah–Singer index theorem #Equivariant map #Expression (computer science) #Fixed point #Fixed-point index #Fixed-point theorem #Geometry #Homotopy and Cohomology in Algebraic Topology #Index (typography) #Mathematical analysis #Mathematics #Pure mathematics #Torus #math.DG #math.KT #math.OA #math.RT
paper · pdf · doi:10.2140/akt.2018.3.235
published as Ann. K-Th. 3 (2018) 235-286 · 51 pages, result for the index pairing added
arxiv created 2016/04/01 · openalex publication_date 2018/03/24 · arxiv updated 2018/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.