The Divisional commander shares his leadership trait with the brigades under his command. Leadership, of course, is extremely important as it determines the efficiency of the unit. For example, a brigade with 26 leadership and 35 efficiency will operate as if its efficiency is 26. So it is a cap, if you will.
Clicking on the info button at the top right of the unit icon will show the units stats during a fight. You will notice that if the brigade has lost its brigadier, there will be a large drop in leadership (it will be shown in red). The remaining leadership is provided by the Divisional Commander. If the Divisional Commander is wounded or killed, all units in the division will suffer a loss in leadership.
If you lose your Divisional Commander and a Brigadier is also lost, that brigades leadership will drop to 0 (or close to it I believe)... and it will be a really bad day for them.
LAVA's post in Divisional Commanders was marked as the answer