Hkr Characters and Higher Twisted Sectors

This is an expository talk, presented at the ChengDu (Sichuan) ICM Satellite conference on stringy orbifolds. It is intended as an introduction to the work of Hopkins, Kuhn, and Ravenel on generalized group characters, which seems to fit very well with the theory of what physicists call higher twisted sectors in the theory of orbifolds. I would like to acknowledge many conversations with Matthew Ando about the contents of this paper. In a better world, he would be its coauthor. 1. Basic definitions 1.0 Since this paper is intended to be expository, I will work in a convenient ad hoc category of orbispaces. For our purposes, an orbispace is a (topological) category X := [X/G] defined by an action of a compact Lie group G on a topological space X , subject to the restriction that the isotropy group Gx of any point x ∈ X be finite. Morphisms of orbispaces are to be equivalence classes, up to natural transformations, of (continuous) functors between categories. This class of objects is rich enough to contain some interesting examples: Ex 1 If M is a reduced d-dimensional orbifold, then its principal orthogonal frame ‘bundle’ O(M) is a smooth manifold upon which the orthogonal group O(d) acts with finite isotropy. By a fundamental lemma of Kawasaki (conceivably known to Satake?) the category (or groupoid) [O(M)/O(d)] is equivalent to the category defined by the original orbifold M. Ex 2 If G is a finite group, then the category [∗/G] with one object, and the set G of morphisms, is an interesting unreduced orbifold. Remarks: Useful topological constructions take us out of the category of smooth objects, so it is convenient to work with a class slightly larger than the usual orbifolds. In general, I will use the mathcal typeface for an orbispace, and the usual mathematical typeface for its underlying space of objects; thus X := [X/G] has objects X and underlying quotient space X/G. However, there will be exceptions: 1.1 If G is a group, and X ∈ (G− spaces), then I(X) := {(g, x) ∈ G×X | gx = x} is itself a G-space, with action defined by h(g, x) = (hgh, hx) . Date: 20 August 2002. 1991 Mathematics Subject Classification. 19Lxx, 55Nxx, 57Rxx. The author was supported in part by the NSF. 1