In the framework of the Compact Light XLS project [1], a Ka-band linearizer with electric field ranging from 100 to 150 MV/m is requested [2, 3, 4]. In order to feed this structure, a proper Ka-band high power klystron amplifier with a high efficiency is needed. This paper reports a possible solution for a klystron amplifier operating on the TM010 mode at 36 GHz, the third harmonic of the 12 GHz linac frequency, with an efficiency of 44% and 10.6 MW radiofrequency output power. We discuss also here the high-power DC gun with the related magnetic focusing system, the RF beam dynamics and finally the multiphysics analysis of a high-power microwave window for a Ka-band klystron providing 16 MW of peak power.