Motivic Pontryagin classes and hyperbolic orientations

We introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups $GL$, ${SL}^c$, $SL$, $Sp$). We show that hyperbolic orientations of $\eta$-periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that $GL$-orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that $\eta$-periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space ${BGL}_n$. Finally, we construct the universal hyperbolically oriented $\eta$-periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum $MGL$.