We investigate the local viscosity of a polymer glass around its glass transition temperature by using environment-sensitive fluorescent molecular rotors embedded in the polymer matrix. The fluorescence of the rotors depends on the local viscosity, and measuring the fluorescence intensity and lifetime of the probe therefore allows us to measure the local free volume in the polymer glass when going through the glass transition. This also allows us to study the local viscosity and free volume when the polymer film is put under an external stress. We find that the film does not flow homogeneously but undergoes shear banding that is visible as a spatially varying free volume and viscosity.