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?