  Colloidal semiconductor nanoplatelets exhibit quantum size effects due to
their thickness of only few monolayers, together with strong optical band-edge
transitions facilitated by large lateral extensions. In this article we
demonstrate room temperature strong coupling of the light and heavy hole
exciton transitions of CdSe nanoplatelets with the photonic modes of an open
planar microcavity. Vacuum Rabi splittings of $66 \pm 1$ meV and $58 \pm 1$ meV
are observed for the heavy and light hole excitons respectively, together with
a polariton-mediated hybridisation of both transitions. By measuring the
concentration of platelets in the film we compute the transition dipole moment
of a nanoplatelet exciton to be $\mu = (575 \pm 110)$ D. The large oscillator
strength and fluorescence quantum yield of semiconductor nanoplatelets provide
a perspective towards novel photonic devices, combining polaritonic and
spinoptronic effects.
