TITLE: Integration in equivariant K-theory
QUESTION [9 upvotes]: Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the compactly supported K-theory of X x W to that of X. It can be represented on the level of classifying maps by regarding a map f from X x W into F as a map from X into the nth loop space of F.
Does a similar picture exist in equivariant K-theory, where X is a G-space and W is a group representation? Of course it does if W is simply a trivial representation. But what if W is some other representation? Is there an integration map, and can one represent it on the level of classifying maps? Is there a good reference for this?

REPLY [7 votes]: Atiyah gave a very precise answer to this question: an equivariant vector bundle V/M is K-orientable (satisfies the Thom isomorphism for K-theory, i.e. K(B(V),S(V)) is a rank-one free module over K(M)) if and only if V has an equivariant Spin_c-structure. Atiyah's proof definitely involves analysis (the family index). The proof by Kasparov is a little more technical, since the definition of the product in Kasparov's KK theory is rather technical. (It is a generalization of the family index theorem of Atiyah and Singer.) But all of these proofs come down to the same thing: the study of families of equivariant Fredholm operators.