A fixed point theorem on noncompact manifolds

TL;DR

This paper extends the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds using KK-theory, providing explicit formulas for equivariant indices and characters of infinite-dimensional representations.

Contribution

It generalizes fixed point formulas to noncompact manifolds and introduces new index formulas involving deformations of principal symbols.

Findings

Fixed point formula for equivariant index on noncompact manifolds

Characters of infinite-dimensional representations realized via index

Connections between deformed symbols and index theory

Abstract

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 non-localised 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.