Quaternionic algebraic cycles and reality

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic K-theory, and the other one classifies an equivariant cohomology theory Z3*(-) which is a natural recipient of characteristic classes KH*(X)-->3*(X) for quaternionic bundles over Real spaces X.