In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the vector bundle over BG associated to V, but the stabilization maps come from the Thom space of (V plus a copies of the regular representation) instead of (V plus a trivial bundle).

The Thom space of (V plus regular representation) isn't of the form Thom(V) smash a sphere, so the details of building the spectrum are complicated.

Two questions:

Is BG^-V definitely different from the more naive Thom spectrum you could build out of the vector bundle V, where you stabilize using the V+ trivial bundles?

What are some other places to read about this spectrum?