  We derive analytic expressions for the leading-order corrections to the
polarization induced in the cosmic microwave background (CMB) due to scattering
off hot electrons in galaxy clusters along the line of sight. For a thermal
distribution of electrons with a kinetic temperature of 10 keV and a bulk
peculiar velocity of 1000 km/s, the dominant corrections to the polarization
induced by the primordial CMB quadrupole and the cluster peculiar velocity
arise from electron thermal motion and are at the level of 10 per cent in each
case, near the peak of the polarization signal. When more sensitive
measurements become feasible, these effects will be significant for the
determination of transverse peculiar velocities, and the value of the CMB
quadrupole at the cluster redshift, via the cluster polarization route.
