  We analyse theoretically the conductivity of a quantum Hall system exposed to
microwave radiation. We find that whenever microwave frequency and cyclotron
frequency are commensurate, there is a {\em resonance} in the longitudinal
conductivity. This resonance has the form of the derivative of a Lorentz
function; precisely at the center of the resonance, the microwave induced
conductivity vanishes. Between the resonances there are maxima and minima, the
depths and precise positions of which depend on the microwave amplitude and the
scattering rate of the impurities. We demonstrate the existence of these
resonances by a microscopic, analytical calculation of the conductivity in
lowest order in the microwave intensity and show here that the conductivity is
independent of the microwave polarization, linear or circular. We then discuss
the general case and predict minima in the longitudinal conductivity
corresponding to fractional values of the microwave frequency divided by the
cyclotron frequency.
