Cyclic cohomology of algebras of smooth functions on orbifolds

Summary

In this informal report I will present several principles that can be used to compute the cyclic cohomology of various smooth algebras. These include

1. (a) C^∞(X), with X a smooth manifold (but by a different method to that of Alain Connes)

2. (b) C^∞(X) ⋊ G, where G is a finite group acting by diffeomorphisms on X

3. (c) C^∞(X)^G or more generally smooth functions on an orbifold

4. (d) C^∞(X), where X is a smooth manifold with boundary or even corners

5. (e) S(G), the convolution algebra of Schwartz functions on a reductive Lie group, including in particular the case G = ℝⁿ.

This last example was in fact the main motivation for this work since at the time it was done (in 1984) I was working on a conjecture of Connes concerning a generalisation of the Connes–Moscovici Index Theorem (see [7]). Here one was interested in computing the pairing between certain specific elements of the cyclic cohomology and the K–theory of S(G). The cyclic cocycles were defined by group cocycles, or equivalently by invariant forms on the homogeneous space G/K via the van Est isomorphism; while the elements of K–theory were represented by abstract indices of twisted Dirac operators, exactly as in [20]. The rough scheme of the computation was to find a concrete ‘heat kernel’ formula for a projection representing the abstract index, substitute it into the cocycle formula and then do a Getzler rescaling to obtain the answer in the limit as Planck's constant approached zero.