A twisted invariant Paley-Wiener theorem for real reductive groups

Let G(+) be the group of real points of a possibly disconnected linear reductive algebraic group defined over R which is generated by the real points of a connected component G'. Let K be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We characterize the space of maps pi bar right arrow tr(pi (f)), where pi is an irreducible tempered representation of G(+) and f varies over the space of smooth, compactly supported functions on G' which are left and right K-finite. This work is motivated by applications to the twisted Arthur-Selberg trace formula.