A high-pressure discharge lamp having a support structure for supporting a light emission tube so as to restrict its displacement in a direction perpendicular to the axis line thereof. A pair of thermal-stress generation members generates thermal stresses due to a temperature change at a time of switching the high-pressure discharge lamp from an on status to an off status. The thermal stresses acts as forces directed downward in a vertical direction and outward with respect to the light emission tube on side tube portions of the light emission tube arranged in a posture where the axis line extends in a horizontal direction.