'act' in this context means, 'he did something, took some action of some kind' but is very vague.
If there is a crisis in the country, or internationally, and someone says, 'The Prime Minister should act', it is implied that the Prime Minister should work with his ministers to deal with the crisis.
In your sentence, it is a scholar. What did he do? Did he teach? Did he write scholarly books?
It is too vague a term for use here.