adverb
under the influence of
Used to indicate that someone or something is under the control, sway, or influence of a person, group, or force. Often appears in formal or written contexts.
彼はその政治家の勢力下にあった。
He was under the influence of that politician.
その地域は長く外国の勢力下に置かれていた。
The region had long been placed under the influence of a foreign power.
Compound of 勢力 (power, influence) + 下 (under) + に (adverbial particle). The exact historical derivation is straightforward from these components.