How does a Hawk jet avoid pitching down when its air brake is applied
It did generate a nose down pitch moment, but the tail had enough control authority to deal with it in most cases. Airbrake extension at greater than 450kt produced an unacceptable trim change. 

Efforts to increase control authority to counter "phantom dive" caused by tail stall during simultaneous flap and gear extension helped. The solution was to affix a leading edge extension to the tail, but since the tail was all moving it was felt that the structure would not be able to handle the extra load. Instead, the tail LEX was affixed to the fuselage and still gave acceptable results. 



A later version, the Goshawk, developed as a military trainer for the US Navy had side mounted airbrakes, though this was done to facilitate the installation of an arrestor hook. This photo shows the LEX just above the side mounted airbrakes.



Hawk mk100 with ventral fins, center airbrake and tail canard LEX.